Does transubstantiation apply to the Last Supper, which occurred prior to the crucifixion?

Yes, this is the teaching of the Catholic Church.

The Catechism of the Catholic Church states:

1366 The Eucharist is thus a sacrifice because it re-presents (makes present) the sacrifice of the cross, because it is its memorial and because it applies its fruit:

[Christ], our Lord and God, was once and for all to offer himself to God the Father by his death on the altar of the cross, to accomplish there an everlasting redemption. But because his priesthood was not to end with his death, at the Last Supper "on the night when he was betrayed," [he wanted] to leave to his beloved spouse the Church a visible sacrifice (as the nature of man demands) by which the bloody sacrifice which he was to accomplish once for all on the cross would be re-presented, its memory perpetuated until the end of the world, and its salutary power be applied to the forgiveness of the sins we daily commit.

The Eucharist according to Catholicism is not just a memorial or symbol, as many Protestants believe, but that the sacrifice of Christ on the cross is truly and actually made present; it is not merely in the past.

In fact, the doctrines concerning the Eucharist seem to be known because of the offering at the Last Supper; if it had not been a real sacrifice, the teachings concerning transubstantiation would likely never have existed:

On that occasion He likewise made an unbloody food-offering, only that, as Antitype, He accomplished something more than a mere oblation of bread and wine, namely the sacrifice of His Body and Blood under the mere forms of bread and wine. Otherwise, the shadows cast before by the "good things to come" would have been more perfect than the things themselves, and the antitype at any rate no richer in reality than the type. Since the Mass is nothing else than a continual repetition, commanded by Christ Himself, of the Sacrifice accomplished at the Last Supper...

Matthew 26:28 For this is My blood of the new covenant, which is shed for many for the remission of sins.
(also see Mark 14:24, Luke 22:20)

And again, Catholic Encyclopedia says concerning the Real Presence of Christ:

The Church's Magna Charta, however, are the words of Institution, "This is my body — this is my blood", whose literal meaning she has uninterruptedly adhered to from the earliest times. The Real Presence is evinced, positively, by showing the necessity of the literal sense of these words, and negatively, by refuting the figurative interpretations.
... [they did not claim the bread and blood to be signs of Christ's body, but] in the original text corpus (body) and sanguis (blood) are followed by significant appositional additions, the Body being designated as "given for you" and the Blood as "shed for you [many]"; hence the Body given to the Apostles was the self same Body that was crucified on Good Friday, and the Chalice drunk by them, the self same Blood that was shed on the Cross for our sins, Therefore the above-mentioned appositional phrases directly exclude every possibility of a figurative interpretation.

The Catholic teaching is that Christ did not claim the body and bread to be a metaphor for His body and blood; He did not say, "this bread is my body," but said "this is my body." He was crucified once for all time -- past, present, and future. Just as He provides salvation to people who lived prior to His sacrifice, so He can be present in the institution of the Eucharist prior to His crucifixion.

... so, when He said: "Take ye, and eat. This is my body", the Apostles received from the hand of the Lord His Sacred Body, which was already objectively present and did not first become so in the act of partaking. This non-dependence of the Real Presence upon the actual reception is manifested very clearly in the case of the Chalice, when Christ said: "Drink ye all of this. For [enim] this is my Blood." Here the act of drinking is evidently neither the cause nor the conditio sine qua non [necessary precondition] for the presence of Christ's Blood.