- Jadon Sancho (39')
- Bruno Fernandes (55')
- Kenny Tete (19')
Fulham took the lead in the 19th minute when Kenny Tete angled in a near post header from a Willian corner, but the London club missed an opportunity to double their advantage seven minutes later when Aleksandar Mitrovic's spot kick was saved by David de Gea.
United took the initiative after the massive let-off and were rewarded in the 39th minute when Sancho equalised with a simple close-range shot, before Fred's defence-splitting pass set up Fernandes to score a 55th-minute winner with a delicate chip.
The result made no change to the Premier League table. Erik ten Hag's side finished the season in third place, while Fulham ended in 10th for their first top-half finish since 2012.
Cairney came within inches of levelling the game in the 76th minute after he blazed an attempt over the cross bar, before an unmarked Wout Weghorst missed a gilt-edged opportunity to end his loan spell in Manchester with a goal.
United will now shift their attention to next week's FA Cup final, in which they will face rivals Manchester City.
Ten Hag's third-place United will be looking to win a second trophy of the season, adding to their Carabao Cup success in February.
"There is still one game to go so, and I am sure that these players will give everything to beat Manchester City next week. So we need it from you as well," Ten Hag told fans at Old Trafford.
"Thank you for everything. I am sure we have a really good chance to take the cup back to Old Trafford."
Fulham manager Marco Silva said his side did not deserve to lose the game, which hinged on the penalty save.
"The game changed completely from that moment," Silva told the BBC. "We should have reacted in a different way. It was a moment for us to show maturity and handle the game in a different way but we haven't done that."
- Robert Jones
|Brighton & Hove Albion||6||5||0||1||+10||15|
|West Ham United||6||3||1||2||+1||10|