Brazilian Rafael missed the decisive spot kick after two goals from Phil Bardsley and Javier Hernandez - and a David de Gea howler - in the dying moments of extra time had made the score 3-3 on aggregate following Sunderland's 2-1 victory in the first leg.
United's Danny Welbeck, Adnan Januzaj and Phil Jones also missed penalties in a remarkable shootout as Sunderland set up a final at Wembley against Manchester City on March 2.
De Gea had earlier put Sunderland within touching distance of the final when he failed to stop a routine shot from Bardsley in the 119th minute and could only watch the ball squirm into the corner of the net.
United hit back in the final minute of extra time, Hernandez sweeping home off the underside of the bar from Januzaj's low cross to force the penalty shootout.
Jonny Evans had given United a first-half lead with a header that looked like it would be enough to send the champions through on away goals until Bardsley popped up and De Gea endured another nightmare.
