I cannot find the exact quote of Augustine. Here are a couple of other patristic texts that I have found:
Pope Julius I
"[The] judgment [concerning Athanasius] ought to have been made, not as it was, but according to the ecclesiastical canon. It behooved all of you to write us so that the justice of it might be seen as emanating from all. ... Are you ignorant that the custom has been to write first to us and then for a just decision to be passed from this place [Rome]? If, then, any such suspicion rested upon the bishop there [Athanasius of Alexandria], notice of it ought to have been written to the church here. But now, after having done as they pleased, they want to obtain our concurrence, although we never condemned him. Not thus are the constitutions of Paul, not thus the traditions of the Fathers. This is another form of procedure, and a novel practice. ... What I write about this is for the common good. For what we have heard from the blessed apostle Peter, these things I signify to you" (Letter on Behalf of Athanasius [A.D. 341], in Athanasius, Apology Against the Arians 20Ã¢â‚¬â€œ35).
Council of Sardica
And this case likewise is to be provided for, that if in any province a bishop has some matter against his brother and fellow-bishop, neither of the two should call in as arbiters bishops from another province.
But if perchance sentence be given against a bishop in any matter and he supposes his case to be not unsound but good, in order that the question may be reopened, let us, if it seem good to your charity, honour the memory of Peter the Apostle, and let those who gave judgment write to Julius, the bishop of Rome, so that, if necessary, the case may be retried by the bishops of the neighbouring provinces and let him appoint arbiters; but if it cannot be shown that his case is of such a sort as to need a new trial, let the judgment once given not be annulled, but stand good as before. (Canon 3)
If it seems good to you, it is necessary to add to this decision full of sincere charity which thou hast pronounced, that if any bishop be deposed by the sentence of these neighbouring bishops, and assert that he has fresh matter in defence, a new bishop be not settled in his see, unless the bishop of Rome judge and render a decision as to this. (Canon 4)
The interesting thing here is that these seem to assert Rome as the court of appeal for disputes between bishops, but not any sort of general executive authority. It's almost as though the Papal See was the "Supreme Court" of the Christian Church at that period of time, or at least that there was a movement or a body of opinion out there that would have it made such. Is it then implicit that Peter has legislative and executive authority, though, to take this governmental metaphor a little further? I'm not sure it is, though I'd be interested in seeing someone reason that out and present a case for it if they believe that it is in fact implicit.
If, however, we say that Rome functioned as simply the ultimate court of appeals in the early church, though, what would that mean for us today? On the one hand, you would have the Roman Catholic Church, which sided with a symbol of unity and ancient judicial authority, thus giving them claim to the "masthead" of Christianity and also to have "legal" backing in some respects, but it would also mean that they'd have severely overstepped their bounds in terms of exercise of authority, which might even be a heresy depending on how far one would want to take it -- an usurpation of the proper roles of the councils or local bishops in terms of legislative and executive authority, and the requirement that communicants believe it to be true as a matter of faith. Meanwhile, the Eastern Orthodox would have preserved the ancient system, but minus any sort of judiciary and without Peter, and without the will to exercise any sort of legislative and executive authority (Arguably a duty of the church, given that new issues arise as time moves on). So, where would such a scenario, if true, lead one to conclude? Are there two pieces of the one true church? Or is there another way to read it?