Palace sensationally reached the final after a 4-3 win over Liverpool, United had reached Wembley in unconvincing fashion having came close to elimination in the third round against Nottingham Forest.
Gary O'Reilly headed the Eagles ahead from a corner but United managed to equalise through Bryan Robson ten minutes before the break.
Just after the hour mark United took the lead, Mark Hughes burying from eight yards.
Substitute Ian Wright levelled the game for Palace, before Robson turned the ball onto the post in the 90th minute but the game went to extra-time.
Wright was once again the man of the moment, the England striker outstretched Steve Bruce and volleyed in to put Palace ahead.
With seven minutes left Danny Wallace set up Hughes to put the ball past Nigel Martyn to equalise.
The game ended in a draw and forced a replay, the second part of the final was significantly less exciting, United winning a bad-tempered encounter 1-0.