United had Chris Smalling sent off after 105 minutes but Lingard popped up five minutes later to score the winner.
The game had been locked at 1-1 after 90 minutes with Jason Puncheon coming off the bench to put London side Palace ahead in the 78th before Juan Mata equalised three minutes later.
United had Chris Smalling sent off after 105 minutes but Lingard popped up five minutes later to give manager Louis van Gaal a first trophy in his two years at Old Trafford.
"It's fantastic to win this title for the club, the fans and also for me," said the Dutchman. "Now I have won a Cup in four countries. Not many managers can say that so I am very happy."
Underdogs Palace had taken the lead when Puncheon, who had only come on six minutes earlier, deftly controlled a long ball into the area and fired past goalkeeper David de Gea.
But within three minutes Wayne Rooney went on a brilliant run across the edge of the Palace area and crossed for Marouane Fellaini to set up the equaliser for Spaniard Mata.
Playing the second period of extra time with 10 men after Smalling's dismissal for a second yellow card, United won the game with Lingard's fierce drive 10 minutes from the end.