Catholicism does not necessarily believe that there was a single moment at which Christ redeemed us:
Christ’s whole life is a mystery of redemption. Redemption comes to us above all through the blood of his cross [cf. Eph 1:7; Col 1:13–14; 1 Pet 1:18–19], but this mystery is at work throughout Christ’s entire life:
already in his Incarnation through which by becoming poor he enriches us with his poverty [cf. 2 Cor 8:9];
in his hidden life which by his submission atones for our disobedience [cf. Luke 2:51];
in his word which purifies its hearers [cf. John 15:3];
in his healings and exorcisms by which "he took our infirmities and bore our diseases" [cf. Mt 8:17; Isa 53:4];
and in his Resurrection by which he justifies us [cf. Rom. 4:25].
(Catechism of the Catholic Church, paragraph 517; all Scripture references taken from footnotes to the passage)
However, the particular aspect of Christ's redemption of us which has to do with our ransom appears to have been fulfilled, that is completed, at the moment of his death:
The redemption won by Christ consists in this, that he came "to give his life as a ransom for many" (Mt 20:28), that is, he "loved [his own] to the end" (Jn 13:1), so that they might be "ransomed from the futile ways inherited from [their] fathers" (1 Pet 1:18).
By his loving obedience to the Father, "unto death, even death on a cross" (Phil 2:8), Jesus fulfills the atoning mission (cf. Isa 53:10) of the suffering Servant, who will "make many righteous; and he shall bear their iniquities" (Isa 53:11; cf. Rom 5:19).
(Catechism, paragraphs 622–623)
Along similar lines, Aquinas' Catena Aurea quotes Origen on the passage:
His ministry extended so far, that He fulfilled even in what follows, And to give his life a ransom for many, they, that is, who believed on Him; and gave it, i.e. to death. But since He was alone free among the dead, and mightier than the power of death, He has set free from death all who were willing to follow Him.
Thus it appears that our redemption, considered as a whole, is caught up with the whole of Christ's life and death; but his death is primary in ransoming us; so that it is at the moment of his death that the ransom was fulfilled.