Forward Wilfried Zaha scored against his former team in the first half as Palace celebrated their first-ever Premier League victory over United at home.
Ralf Rangnick's tenure as interim manager came to an end with United collecting 58 points, their lowest tally in Premier League history.
United were nevertheless able to hold on to the sixth place after West Ham United suffered a 3-1 defeat at Brighton & Hove Albion and had to settle for seventh.
Watched by new manager Erik ten Hag from the stands, United produced a lacklustre display and went behind in the 37th minute as Zaha finished smartly from the edge of the box after picking up a loose pass and cutting across the United defence.
The visitors tried to improve their game in the second half but they looked poor in attack and barely created chances as Patrick Vieira's Palace ended the season on a winning note and took the 12th spot.