The Manchester United loanee broke the deadlock 13 minutes after half-time, converting Lucas Digne's cross after a crisp Villa move.
Five minutes later, Morgan Rogers was fouled inside the box and Rashford stepped up to stroke home his second from 12 yards.
Soon enough, the game was completely killed off when Jacob Ramsey turned on the afterburners to race through midfield and slam home a third.
Preston had been competitive in the first hour, Stefan Thordurson heading a presentable chance wide in the first half, while their Premier League visitors were limited to chances from set-pieces.
Suffering a long list of absences, however, North End could not keep up the pace and it is Unai Emery's side on their way to Wembley.