Jesse Lingard's brilliant volley in the 111th minute gave United their first FA Cup triumph title since 2004 that matched Arsenal's record mark.
United finished with 10 men after Chris Smalling was sent off in extra-time for a second yellow card.
Jason Puncheon had given Palace, seeking a first major honour, the lead in the 78th minute only for Juan Mata to equalise just three minutes later.
- DPA