After a 10th-minute goal from Bruno Fernandes, which was netted thanks to a deflection from Nat Phillips, Liverpool needed to come from behind to earn their first win at Old Trafford since 2014.
Jota scored the equalizer for the Reds in the 34th minute before Liverpool taking the lead right before the half-time whistle with a goal from Roberto Firmino.
The Brazilian doubled Liverpool’s lead over their arch-rival in the 47th-minute before Marcus Rashford gave comeback hopes to the Red Devils in the 68th-minute.
However, a late goal from Mohamed Salah sealed Liverpool a 4-2 away Premier League victory over Manchester United on Thursday.
Liverpool’s manager, Jurgen Klopp, told Sky Sports: "We reacted really well on their goal, it was unlucky in that moment but it was a bit like we started the game to be honest.
"The defending in that moment, letting Bruno, half right side, coming inside - there was nobody.
"He could have used his left, could've used the outside of his right foot and it was unlucky in the centre but the defending in that moment wasn't good.
"They overloaded the wings obviously and we didn't adapt well to that in the first 15 minutes or so.
"Then we got control and from that moment on we could play what we want to play. That was absolutely okay and second half even better.
"Great fight, good game, intense, how a derby should be, and since I'm at Liverpool the first win at Old Trafford.
"Good timing, I would say! We needed it, we fought hard for it, we deserved it and so all good."
On the incident between himself and Sadio Mane, who refused to acknowledge his manager when he was leaving the pitch, he said: "There's no problem.
"Yesterday I made a late decision in training to decide Diogo. I explain things usually but there was no time for that.
"That's all. It's all fine."
Chelsea's Wednesday defeat to Arsenal opened the door for Liverpool to claim a top-four finish in their poor title-defending season.
However, Klopp expects his side’s next three games will be extra challenging for the reds.
"We are still in the race, that's all that we could have done tonight, we did that and now we have a very intense period playing the next three games in a week.
"In three days West Brom, Burnley with supporters, and then Crystal Palace at home, which we are looking forward to for different reasons because our supporters are there then hopefully.
"We have to recover quickly and to go again."