As @Hugh said, To "obligate" in Modern Greek is υποχρεώνω, Ancient ὑποχρεόω < ὑπόχρεως. In Modern Greek the verb has come to mean pretty much "to force someone to do something", but the notion of indebtedness is still there: με υποχρεώνεις is still used in the sense of "I am in your debt, I feel obliged to you". So semantically, this is exactly what OP is after.
The catch is that, while ὑπόχρεως is used for "indebted" in Aristophanes, the meaning "bound, obliged" according to LSJ is Koine—Plutarch, Dionysius of Halicarnassus, Polybius; and the verb has no Classical or, as far as I can tell, Byzantine usage at all. Triantafyllidis' Modern Greek dictionary calls this a learnèd modern coinage after French obliger, and that appears to be right.
"oblige" does turn up as a verb in LSJ, but the Greek for it is not about the obligation of debt, but about the implicit obligation in a favour: εὐχαριστέω "bestow a favour on, oblige" (only inscriptions in that meaning), χαρίζω "say or do something agreeable to a person, show him favour or kindness, oblige, gratify", χάριν φέρω "confer a favour on one, do a thing to oblige him". Of course, "oblige" in English can mean just "gratify" without an expectation of reciprocation, so maybe LSJ's gloss is misleading. (After all, χάρις in Christian theology is "Grace", which is by definition unreciprocated. People are more complicated than that, hence the implicature.)
Not finding anything promising in Philebus, apart from one instance of οἷς δεῖ χάριν ἔχειν ("There are some clever people who try to prove this theory to us, and we ought to be grateful to them.") But if you don't think Polybius is too late, his usage of ὑπόχρεως is what you describe:
ὁρῶντες γὰρ τὸ συνέχον τοῖς Βυζαντίοις τῆς ὑπομονῆς τοῦ πολέμου κείμενον ἐν ταῖς κατὰ τὸν Ἀχαιὸν ἐλπίσι, θεωροῦντες δὲ τὸν πατέρα τὸν Ἀχαιοῦ κατεχόμενον ἐν Ἀλεξανδρείᾳ, τὸν δ’ Ἀχαιὸν περὶ πλείστου ποιούμενον τὴν τοῦ πατρὸς σωτηρίαν, ἐπεβάλοντο πρεσβεύειν πρὸς τὸν Πτολεμαῖον καὶ παραιτεῖσθαι τὸν Ἀνδρόμαχον, καὶ πρότερον μὲν ἐκ παρέργου τοῦτο πεποιηκότες, τότε δ’ ἀληθινῶς σπεύδοντες ὑπὲρ τοῦ πράγματος, ἵνα προσενεγκάμενοι πρὸς τὸν Ἀχαιὸν τὴν χάριν ταύτην ὑπόχρεων αὐτὸν ποιήσωνται πρὸς πᾶν τὸ παρακαλούμενον.
For observing that the chief cause of the Byzantines' resolute endurance of the war lay in their hopes of support from Achaeus, and knowing that Achaeus' father was a prisoner at Alexandria and that Achaeus above all things desired his deliverance, they decided to send an embassy to Ptolemy begging him to give up Andromachus. They had indeed previously made this request without insisting much on it, the now they pressed it most seriously, in order that by doing this favour to Achaeus they might put him under such an obligation that he would do all they demanded. (Polybius 4.51.2)