Frank Lampard’s side lost 1-0 at home to Leeds United, but pulled off a thrilling 4-2 victory at Elland Road on Wednesday, to go through with a 4-3 aggregate.
Villa qualified for the final, after seeing off Midland rivals West Brom on Tuesday night.
Craig Dawson scored the only goal at Hawthorns to level on aggregate and send the tie all the way to penalties.
Jed Steer then saved two penalties for Villa, while Tammy Abraham converted the final kick to send his side to Wembley for the second consecutive season.
The winner will gain an automatic spot in the Premier League, along with Norwich and Sheffield United.