Two goals from Nigerian Calvin Bassey and Alex Iwobi were enough to give Fulham a historic away at Old Trafford for 21 years.
The win at Old Trafford is Fulham's second at the United stadium since October 2003.
Bassey fired the opener in the second half before United's defender Harry Maguire levelled in the 80th minute.
However, the match turned into a shellshock around Old Trafford - just as you felt they were going to win with heightened attacks on the Fulham defence Iwobi restored the Londoner's lead in the dying minutes of added time.
The result sees fourth-placed Villa open and an eight-point gap to sixth-placed Manchester United.
United were without injured striker Rasmus Hojlund, making them rather ineffective in attack until the final Maguire equaliser seemed to have rescued a point for the hosts.
The win now means Fulham has ended their 11 away games without a win in the Premier League this season.
The matchwinner Iwobi said he was over the moon after the historic victory.
"We feel we performed well and have been performing well recently, it is just about getting the results," Iwobi said.
