It seems that many ancient philosophers made this claim or a related one, but was Socrates the first? Did any of the pre-Socratics make any similar claims? Was this a common or novel idea at the time Socrates said it?
There is no reason to think that Socrates thought this, but if he did, then yes, he was the first.
You might have the impression that Socrates claimed this because of things that Plato puts in his mouth in his dialogues. It could be the case that Plato was accurately representing Socrates' views, but if so, there is no external evidence for this interpretation of Plato. See my answer here for more:
None of our testimonies of Socrates [the historical person] attribute to him the belief that virtue is sufficient for happiness, but it is clear that Socrates did have an interest in the definitions of virtues: "Socrates was busying himself about ethical matters and neglecting the world of nature as a whole but seeking the universal in these ethical matters, and fixed thought for the first time on definitions" (Aristotle, Metaphysics I.6).
But Aristotle doesn't even say that Socrates ever had an opinion on the definitions of the virtues, let alone whether any or all the virtues were sufficient for happiness.
However, let's say that Plato does accurately represent Socrates' beliefs. What then?
None of the so-called Socratic dialogues, which some scholars do hypothesize represent Socrates' beliefs (see the link above for more details), ever come out and say that virtue is sufficient for happiness. What is misleading is that Socrates in these dialogues does talk about the supreme importance of virtue (e.g., in the Apology), but this is consistent with the view that virtue is merely necessary for happiness.
But some of the later Platonic dialogues do seem to affirm the view that virtue is sufficient for happiness.
For instance, here is the Theaetetus:
SOCRATES: Evils, Theodorus, can never pass away; for there must always remain something which is antagonistic to good. Having no place among the gods in heaven, of necessity they hover around the mortal nature, and this earthly sphere. Wherefore we ought to fly away from earth to heaven as quickly as we can; and to fly away is to become like God, as far as this is possible; and to become like him, is to become holy, just, and wise. But, O my friend, you cannot easily convince mankind that they should pursue virtue or avoid vice, not merely in order that a man may seem to be good, which is the reason given by the world, and in my judgment is only a repetition of an old wives' fable. Whereas, the truth is that God is never in any way unrighteous—he is perfect righteousness; and he of us who is the most righteous is most like him. Herein is seen the true cleverness of a man, and also his nothingness and want of manhood. For to know this is true wisdom and virtue, and ignorance of this is manifest folly and vice. All other kinds of wisdom or cleverness, which seem only, such as the wisdom of politicians, or the wisdom of the arts, are coarse and vulgar. The unrighteous man, or the sayer and doer of unholy things, had far better not be encouraged in the illusion that his roguery is clever; for men glory in their shame—they fancy that they hear others saying of them, 'These are not mere good-for-nothing persons, mere burdens of the earth, but such as men should be who mean to dwell safely in a state.' Let us tell them that they are all the more truly what they do not think they are because they do not know it; for they do not know the penalty of injustice, which above all things they ought to know—not stripes and death, as they suppose, which evil-doers often escape, but a penalty which cannot be escaped.
THEODORUS: What is that?
SOCRATES: There are two patterns eternally set before them; the one supremely happy [εὐδαιμονεστάτου] and divine, the other godless and wretched: but they do not see them, or perceive that in their utter folly and infatuation they are growing like the one and unlike the other, by reason of their evil deeds; and the penalty is, that they lead a life answering to the pattern which they are growing like. And if we tell them, that unless they depart from their cunning, the place of innocence will not receive them after death; and that here on earth, they will live ever in the likeness of their own evil selves, and with evil friends—when they hear this they in their superior cunning will seem to be listening to the talk of idiots (176b-177a).
And here is the Timaeus, and these words are not spoken by Socrates but by Timaeus:
Whoso, then, indulges in lusts or in contentions and devotes himself overmuch thereto must of necessity be filled with opinions that are wholly mortal, and altogether, so far as it is possible to become mortal, fall not short of this in even a small degree, inasmuch as he has made great his mortal part. But he who has seriously devoted himself to learning and to true thoughts, and has exercised these qualities above all his others, must necessarily and inevitably think thoughts that are immortal and divine, if so be that he lays hold on truth, and in so far as it is possible for human nature to partake of immortality, he must fall short thereof in no degree; and inasmuch as he is for ever tending his divine part and duly magnifying that daemon who dwells along with him, he must be supremely happy [εὐδαίμονα]. And the way of tendance of every part by every man is one—namely, to supply each with its own congenial food and motion; and for the divine part within us the congenial motions are the intellections and revolutions of the Universe. These each one of us should follow, rectifying the revolutions within our head, which were distorted at our birth, by learning the harmonies and revolutions of the Universe, and thereby making the part that thinks like unto the object of its thought, in accordance with its original nature, and having achieved this likeness attain finally to that goal of life which is set before men by the gods as the most good both for the present and for the time to come (90b-e).
Note that these are probably the two passages in Plato from which one could make the strongest case that Plato believed in the sufficiency of virtue for happiness, but we might be struck by how little attention is paid overall to stressing this point. Further, in the Philebus, Plato does find a place for other things (such as pleasure) in the good life, which actually takes away from the possibility that virtue is sufficient for happiness.
Why is Plato so unclear? In some places, he sounds like virtue is sufficient. But then in other places, such as the Philebus, he thinks that pleasure is an ingredient of the good life.
It seems to me that the question of whether virtue is sufficient for happiness is not developed in full until the Hellenistic period when the Stoics look at the classical period (perhaps specifically at the Cynic school of philosophy) and begin to develop their own view that virtue is sufficient. It seems that they are inspired by the Cynics, but it is difficult to be certain what the Cynics thought. And Aristotle was famously associated in the Hellenistic period (and afterwards) with the view that virtue is not sufficient for happiness and that we need other things such as bodily and external goods. Perhaps that's a mistaken interpretation of Aristotle but it is very unclear, much like it is with Plato.
I think that the ambiguity of classical philosophers reflects the fact that the relevant philosophical debate did not become articulated precisely until the Hellenistic period, and I think that if we wanted to find this view in classical or even archaic Greek thinkers, we're not going to have as much success as we might want to.