United and Everton were destined for extra time until the 93rd minute when Martial ran through on goal and slotted home to send the red half of Wembley wild.
"The penalty save was a key moment because they were doing really well," De Gea told BBC Sport. "But I thought we played well and I am really happy.
"Anthony Martial is unbelievable and we won the game in the last minute. It was unbelievable.
"It is really important for us, the fans and the club and we now have four games to go in the league and we want to finish in the top four. That is what we want."
The winner of tomorrow's semi-final between Watford and Crystal Palace will take on United in next month's final at Wembley.