Matt Targett opened the scoring for Villa, only for the hosts to be pegged back by Kelechi Iheanacho midway through the second half.
But with the game seemingly destined for penalties, Trezeguet got on the end of a delightful Ahmed Elmohamady cross to fire in the winner and send Villa Park into delirium.
Smith told Sky Sports: “I certainly let myself go with the goal. That deep onto stoppage time, there was a bit of emotion from everyone and the place erupted.
“Our goalkeeper made three great saves but I thought we were aggressive as well.
“To go and get the win at the death like we did there… To take this club to Wembley, I’ll be a very proud man to walk out there with this team.”