The rescheduled match from May 2 once again saw protests outside of Old Trafford against the Glazer family that owns Man United, but the game was able to be played with both teams arriving safely at the stadium.
Liverpool's first victory at Old Trafford in seven years left them knowing that winning their last three games will give them a great chance of breaking into the top four at the end of a disappointing season.
"Great fight, good game, intense, and the first win at Old Trafford since I am here," said Liverpool manager Jurgen Klopp. "Good timing, I would say. We fought hard for it, we deserved it".
Bruno Fernandes put the home side in front on 10 minutes, dispatching from an Aaron Wan-Bissaka pass that deflected into the goal to put further pressure on a Liverpool team chasing the remaining Champions League places.
Liverpool then took the lead just before half-time on the back of a Firmino bullet header from a set piece that saw the hosts go into the dressing room trailing 2-1.
Things got worse for United as the second half started, a pair of bad giveaways led to Henderson spilling the rebound from a Trent Alexander-Arnold shot into Firmino's path for the Brazilian's second goal of the match.
United pushed for an equaliser, but Mohamed Salah put the game to bed just before full-time, calmly slotting past Henderson after a long run down the right flank to seal the points.
The result puts Liverpool on 60 points from 35 games and in fifth place, four behind Chelsea in fourth, but with a game in hand. Man United are still in second on 70 points and look set to stay in the top four with two matches remaining.
Third-placed Leicester face Chelsea next Tuesday, which could provide an opening for Liverpool.
"We are still in the race, that's all we could have done tonight," said Klopp.
Information from Reuters was used in this report.