Ashley Young was the provider of both United goals, the winger winning a first-half penalty when Scott Dann handled his cross.
Juan Mata scored from the spot before Jason Puncheon levelled for Palace with a deflected free-kick after the break.
Fellaini then headed in Young's cross on 78 minutes.
Champions League
The team which finishes fourth in the Premier League will face a play-off to reach the group stages of the Champions League.
Louis van Gaal's side were grateful for some superb saves from David De Gea, but they once again showed character as they recovered from Puncheon's equaliser and a period of sustained Palace pressure to win the game.
Crystal Palace 1-2 Man Utd: Van Gaal says work ethic key to victory
Their remaining rivals for the final Champions League spot are Liverpool, and United will return to the Champions League if Brendan Rodgers' side lose at the home of newly-crowned champions Chelsea on Sunday.
Young had previously set up just two goals this season, but he doubled that tally of contributions at Selhurst Park with an accomplished performance on the left wing.
He presented Mata with the chance to put the visitors in front when his cross was handled by Dann - the Palace defender leaning into the ball as he misjudged its trajectory.
It was the only real highlight of a first half which saw very few chances, but United suffered injuries to both full-back Luke Shaw and lone striker Wayne Rooney which led to the England pair being substituted.
Palace improved after the break, with Puncheon stepping from the bench to bend home a free-kick via a deflection off Daley Blind in the United wall.
Penalty decision was wrong - Pardew
The Eagles had their Premier League safety assured by results before the evening kick-off and, with the pressure off, they played some dynamic football in the final third.
James McArthur poked just wide while De Gea made a point-blank save to deny Glenn Murray as wingers Yannick Bolasie and Puncheon started to dominate the game.
However, it was United winger Young who had the final say when his superb delivery from deep picked out Fellaini, who was surrounded by three markers but in exactly the right place to nod the ball into the empty net.
De Gea made another tremendous save to deny Murray and put United on the verge of a return to Europe's elite competition.
Crystal Palace manager Alan Pardew:
"We were a lot better in the second half and I thought we would go on and get a victory - or at least a draw. It is galling to be talking about a defeat.
"I think the referee's in a bad position to say Scott Dann's offence warrants a penalty - and I told him that. The ball came in, there was no real threat and Scott leans in and is turning away. I think it hits the shoulder and his chest. From his angle the referee couldn't tell - it was impossible. I think it is a really tough call.
"We had a great moment when Glenn Murray had a great chance - it was a fantastic save from one of the best goalkeepers in the world."
Manchester United manager Louis van Gaal:
"We still have to win one point I believe to make sure of Champions League qualification. But we have made a very important step.
"We have worked so hard, the players showed the team spirit is good. I am very proud. When you are fighting like this every week, we are difficult to beat.
"David De Gea has made great saves several times this season, I am very happy he saved from Murray because at the time it could have been 2-1 for Crystal Palace."
Juan Mata scored from the spot before Jason Puncheon levelled for Palace with a deflected free-kick after the break.
Fellaini then headed in Young's cross on 78 minutes.
Champions League
The team which finishes fourth in the Premier League will face a play-off to reach the group stages of the Champions League.
 |
| Ashley Young was the provider of both United goals, the winger winning a first-half penalty when Scott Dann handled his cross. |
Crystal Palace 1-2 Man Utd: Van Gaal says work ethic key to victory
Their remaining rivals for the final Champions League spot are Liverpool, and United will return to the Champions League if Brendan Rodgers' side lose at the home of newly-crowned champions Chelsea on Sunday.
Young had previously set up just two goals this season, but he doubled that tally of contributions at Selhurst Park with an accomplished performance on the left wing.
He presented Mata with the chance to put the visitors in front when his cross was handled by Dann - the Palace defender leaning into the ball as he misjudged its trajectory.
It was the only real highlight of a first half which saw very few chances, but United suffered injuries to both full-back Luke Shaw and lone striker Wayne Rooney which led to the England pair being substituted.
Palace improved after the break, with Puncheon stepping from the bench to bend home a free-kick via a deflection off Daley Blind in the United wall.
Penalty decision was wrong - Pardew
The Eagles had their Premier League safety assured by results before the evening kick-off and, with the pressure off, they played some dynamic football in the final third.
James McArthur poked just wide while De Gea made a point-blank save to deny Glenn Murray as wingers Yannick Bolasie and Puncheon started to dominate the game.
However, it was United winger Young who had the final say when his superb delivery from deep picked out Fellaini, who was surrounded by three markers but in exactly the right place to nod the ball into the empty net.
De Gea made another tremendous save to deny Murray and put United on the verge of a return to Europe's elite competition.
Crystal Palace manager Alan Pardew:
"We were a lot better in the second half and I thought we would go on and get a victory - or at least a draw. It is galling to be talking about a defeat.
"I think the referee's in a bad position to say Scott Dann's offence warrants a penalty - and I told him that. The ball came in, there was no real threat and Scott leans in and is turning away. I think it hits the shoulder and his chest. From his angle the referee couldn't tell - it was impossible. I think it is a really tough call.
"We had a great moment when Glenn Murray had a great chance - it was a fantastic save from one of the best goalkeepers in the world."
Manchester United manager Louis van Gaal:
"We still have to win one point I believe to make sure of Champions League qualification. But we have made a very important step.
"We have worked so hard, the players showed the team spirit is good. I am very proud. When you are fighting like this every week, we are difficult to beat.
"David De Gea has made great saves several times this season, I am very happy he saved from Murray because at the time it could have been 2-1 for Crystal Palace."
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