Let us look at some primary sources, and avoid "riding the pony" of modern commentaries:
The phrase comes from Sermon 131.10 of St. Augustine, the Latin is:

jam enim de hac causa duo concilia missa sunt ad sedem apostolicam; inde etiam rescripta venerunt; causa finita est
[http://www.newadvent.org/cathen/11604a.htm]

Translated, it reads,

. . . for already on this matter two councils have sent to the Apostolic See, whence also rescripts (Rome "spoke") have come. The cause is finished.

Council of Carthage, June 416 AD, To Innocent I:
(Regarding Celestius) We had come according to custom to the Chuch of Carthage, and a synod was held for various affairs, when our fellow-priest Orosius presented us with letters from our holy brothers and fellow-bishops Heros and Lazarus, which we enclose. These having been read, we perceived that Pelagius and Celestius were accused of being authors of a wicked error, which must be anathematized by all of us. Wherefore we asked that all which had been done with regard to Celestius here in Carthage about five years ago should be gone through...This act, lord brother, we thought right to initmate to your holy charity, in order that to the statutes of our littleness might be added the authority of the Apostolic See (ut statutis nostrae mediocritatis etiam apostolicae sedis adhibeatur auctoritas) for the safety of many, and the correction of the perversity of some.
(Chapman, Dom John, Studies on the Early Papacy, 140.)

(Then speaking about Pelagius) 4. If, therefore, Pelagius seems to your holiness to have been justly absolved by the Episcopal acts which as said to have been transacted in the East, at all events the error itself and the impiety, which now has many asserters in different places, ought to be anathematized by the authority of the apostolic see also
(ibid, 141.)

Council of Mileve, 416 A.D., To Innocent I:
Since God has by a special gift of his grace set you in the apostolic see, and has given such a man to our times, so that it could be rather imputed to us as a fault of negligence, if we withheld from your reverence whatever is to be furnished for the Church, than that you should be able to receive the same contempt or neglect, we beseech you to apply your pastoral care to the great peril of the weak members of Christ.

The authors of this most pernicious error are said to be Pelagius and Celestius, whom, indeed, we should prefer to be cured in the Church, rather than that they should be cut off from the Church, if no necessity compels…

We consider that by the help of the mercy of our Lord God, who deigns both to direct your counsel and to hear your prayers, those who hold such perverse and pernicious opinions will more easily yield to the authority of your holiness, drawn from the authority of Holy Scripture, so that we may be congratulated by their correction, than saddened by their ruin…

We write this from the council of Numidia, imitating our colleagues of the church and province of Carthage, who we understand have written on this matter to the apostolic see, which your blessedness adorns.
(ibid, 141-142.)

Aurelius, Alypius, Augustine, Evodius, and Possidius, To Innocent, A.D. 416
We send to your holiness letters from the two councils of the provinces of Carthage and Numidia (Mileve)…

For we have heard that there are some in the city of Rome, where he [Pelagius] lived long, who favored him for various reasons, some clearly because he is said to have persuaded them, but more because they do not believe him to hold such views, especially as it is boasted that ecclesiastical acts were drawn up in the East, where he is living, by which he is declared innocent. If indeed the bishops pronounced him Catholic, we must believe that it was for no other reason than because he said he acknowledged the grace of Christ…

Of the rest of the accusations against him doubtless your beatitude will judge in the same way as the acts of the two Councils. Doubtless you kindness of heart will pardon us for having sent to your Holiness a longer letter than you might perhaps have wished. For we do not pour back our little stream for the purpose of replenishing your great fountain (non enim riuulum nostrum tuo fonti augendo refundimus); but in the great temptation of these times (from which may He deliver us to whom we say, "and lead us not into temptation") we wish it to be approved by you whether our stream though small, flows from the same head of water as your abundant river, and to be consoled by your answer in the common participation of the same grace.
(Ep. 177, vol. ii, pp 764-772, qtd in Chapman, 144-145.)

Pope Innocent replies:
Innocent I, to the Council of Carthage, January 27, 417
In making inquiry with respect to those things that should be treated with all solitude bu bishops, and especially by a true and just Catholic Council, by preserving, as you have done, the example of ancient tradition, and by being mindful of ecclesiastical discipline, you have truly strengthened the vigour of our religion, no less now in consulting us before passing sentence. For you decided that it was proper to refer to our judgment, knowing what is due to the Apostolic See, since all we who are set in this place, desire to follow the Apostle from whom the very episcopate and whole authority of this name is derived. Following in his steps, we know how to condemn the evil and to approve the good. So also, you have by your sacerdotal office preserved the customs of the Fathers, and have not spurned that which they decreed by a divine and not human sentence, that that whatsoever is done, even though it be in distant provinces, should not be ended without being brought to the knowledge of this See, that by its authority the whole just pronouncement should be strengthened, and from it all other Churches (like waters flowing from their natal source and flowing through the different regions of the world, the pure streams of one incorrupt head), should recieve what they enjoin, whom they ought to wash, and whom that water... I congratulate you, therefore, dearest brethren, that you have directed letters caring for the other Churches which you rule, you also show your solicitude for the well-being of all, and you ask for a decree that shall profit all the Churches of the world at once.
(Chapman, 146-147.)

Innocent I, to the Council of Mileve, January 27, 417
...why else did you confirm it with your action, if you were not aware that responses ever flow from the Apostolic fountain to all provinces for those who ask them? Especially as often as a question of the faith is discussed, I think that all our brothers and fellow-bishops should refer to none other than Peter, the author of their name and office, even as now your charity has referred to us a think which may be useful throughout the world to all the Churches in common.
(ibid, 148.)

We declare that Pelagius and Celestius, that is the inventors of new doctrines which, as the apostle said, are wont to produce no edification, but rather utterly empty questionings, should be by the authority of apostolic vigour be deprived of ecclesiastical communion (excommunication), until they recover from the snares of the devil, by whom they are held prisoners according to their own choice; and that meanwhile they should not be received within the Lord’s fold, because, following the course of a crooked way, they have themselves chosen to desert.
(Giles, E. Documents Illustrating Papal Authority, 202-203.)

So, did the "two councils" speak, or did they appeal to the Bishop of Rome for a decision? Clearly, from the primary sources cited above, they didn't "speak," they appealed to "the great fountain" and likened themselves to "little streams" and, according to the "authority of (his) holiness, drawn from the authority of Scripture," they submitted their case to Rome for a decision. Then, as St. Augustine summarized in his Sermon 131, later that year, after the two councils were sent to Rome, Rome spoke, and St. Augustine considered the case closed (with regard to the two Pelagians).

It must also be noted that Pope Innocent's decree was not "final" in regard to the two Pelagians. He said, they "should be by the authority of apostolic vigour be deprived of ecclesiastical communion (excommunication), until they recover from the snares of the devil." So IF they appealled and/or repented of their ways, in the eyes of the Apostolic See, the case is NOT "closed."

So what happened next?
Well, Pelagius and Celestius presented [u]false confessions[/u] to Pope Zosimus (Pope Innocent had died in early 417, shortly after his "ruling" on the Pelagians). Pelagius had also written a confession entitled Libellus Fidei, in 417. E. Giles (the source for the above quotes and an Anglican with no motive to support the cause of the papacy) admits that Pelagius’ confession is orthodox – but includes some evasive passages on free will and baptism (baptism being one of the chief precepts of Pelagianism). (ibid, 207.) From a commentary on the University of Notre Dame web site: “Pope Zosimus, whom the stratagems of Celestius had for a moment deluded…”[http://www.nd.edu/Departments/Maritain/etext/augustin.htm]

The African bishops are still quite sure that the Pelagians still adhere to their heresy, and don't back down. It is NOW that St. Augustine's Sermon 131 comes into existence. As far as St. Augustine is concerned, the two councils were sent to Rome for a decision, Rome had spoken, the case is closed.

In 418, Pope Zosimus is convinced that he had been deceived, and he reverses his statement regarding the orthodoxy of Pelagius and Celestius and renews the excommunication of the heretics. The deception of Zosimus was short-lived, but notorious.

So, in summary - Pelagius and Celestius were notorious Pelagians. The African bishops appealled to the Apostolic See (Rome) for a decision on them. Rome responded (spoke) through Pope Innocent, excommunicating the two - but left the door open for reconcilliation. The two heretics falsely "confessed" to now Pope Zosimus - who reinstated them to communion with the Catholic Church. The African bishops stood firm, knowing the confessions were fallacious, shortly thereafter Pope Zosimus capitulated and reinstated the excommunications.

What we ALSO see is that the "two councils" truly sought Rome's "decision" in the matter. They didn't decide it for themselves, they appealed to Rome - and after Rome spoke, they were content the "case is closed." It was closed, until the two heretics lied to Innocent's successor, Pope Zosimus, who declared they were faithful Catholics, but shortly thereafter retracted that statement and declared them (again) to be heretics..

I believe I have adequately presented, from primary sources, that 1) "the two councils" were submitting to the higher authority, the "great fountain," for a decision on these two Pelagians and the error they taught, so as to benefit the whole Church. 2) That St. Augustine clearly stated, after those two councils submitted themselves to Peter, that Rome had spoken, and to him (St. Augustine) the case was closed.

I have further documentation on the deceptive "confesssions" of Celestius and Pelagius, but since this posting has gotten rather long already, I'll stop for now.

Respectfully yours,
Scott Windsor<<<
Sept. 23, 2003 - Feast of St. Padre Pio
(Posted Sept. 24, 2003)

PS- Giles was Mr. King's earlier source of primary sources, and Giles' source was Chapman. Chapman was the translator and Giles' citations are nearly identical to Chapman's, but Chapman includes a few more details that Giles (an Anglican) leaves out (I have both sources).