Substitute Jason Puncheon had fired beyond David de Gea on 77 minutes to give Palace the advantage as a hitherto low-key match came to life late on.
But United were level within minutes. Marouane Fellaini controlled Wayne Rooney's cross with his chest allowing Juan Mata to volley low past Hennessey.
United captain, Rooney, was lucky to escape punishment for a challenge on Puncheon in the first period of extra time.
However, Chris Smalling wasn't so fortunate as he received a second yellow card followed by his marching orders for hauling down Yannick Bolasie when Palace broke moments later.
Palace immediately looked to take advantage of its numerical advantage, coming close through substitute Dwight Gayle.
Yet it would be another sub who would have the final say. Jesse Lingard ran onto a blocked shot from United striker, Anthony Martial, and let rip a stunning volley that arrowed into Hennessey's top corner.
'A special moment'
"I'm delighted. It's a special moment. It's a competition I have loved watching as a fan growing up," United captain Wayne Rooney told BBC Sport as he celebrated his first ever FA Cup triumph on the pitch after the match.
"I've been to two finals and lost, so I will enjoy it tonight for sure," he added.
United coach, Louis Van Gaal was equally delighted to pick up his first trophy as a manager in England.
"It is fantastic to win this title for the club, for the fans, and also for me because I now have won the cup in four countries, and not many managers have done that," Van Gaal said.
Van Gaal has been under severe pressure to bring silverware and winning football to the Old Trafford club.
It was reported by BBC Sport
and Sky Sports
shortly after the full time whistle that the Dutchman would be replaced as United boss by former Chelsea and Real Madrid manager Jose Mourinho early next week.
Yet when these reports were put to Van Gaal by reporters at the post match press conference, he sharply refused to entertain them.
"I show you the (FA) Cup. I don't discuss (my future) with the media," he said in comments carried by the UK's Press Association.
The Palace wait goes on
For a vanquished Palace, the wait for the club's first major trophy goes on.
The south London club was founded in 1910 and has yet to win one of England's major prizes.
But the Eagles manager, Alan Pardew, remained proud of his players despite the disappointment.
"My players gave everything. Everything. And they deserved to win but the game is like that.
Pardew was part of the last Palace team to reach an FA Cup final back in 1990, when it was defeated by Alex Ferguson's Manchester United after a replay.
"It's such a shame. We did a lot of things very well, we hurt them on the break, the referee didn't help us but we grew into the game.
"I'm so proud of the club, we have been dignified in defeat and we look forward to next year," he added.