The Reds took the lead after a frantic opening 20 minutes as Sadio Mane controlled inside the United box and finished coolly from Fabinho's lofted pass.
United pegged the home side back before the break as Jesse Lingard took advantage of Alisson's error after the Brazilian spilled a cross from Romelu Lukaku.
With 20 minutes left, substitute Xherdan Shaqiri became the hero for the Reds, with his two deflected shots breaking the United barrier to give Liverpool a 3-1 victory.
Liverpool had 36 shots at David de Gea's goal and they looked to be in the mood all night. Speaking after the game, Klopp was delighted with the level of performance from his side.
Asked how happy he was with his players, he said: "Very! Very, very, very. The brilliant start was one of the best performances we had since I'm in Liverpool, to be honest, not only this season.
"The first half an hour was outstanding, outstanding. We scored only once but I don't think anyone expects to score three, four, five times or whatever. How we played was just perfect, really.
"We tried to pass in the right areas, we accelerated, we wanted the ball back, we played in behind the line, stuff like that. It was just good.
"Then after half an hour we lost the momentum of the game a little bit, they scored and it was a bit more open. I had no problem with the half-time whistle, honestly, and then we could adjust a few things.
"It was not that I thought then that Untied was manning the game or whatever, but it was just a bit more open. Then they changed and when Fellaini is on the pitch it is never a good sign for the other team, to be honest, because these balls are really difficult to defend.
"But we did even that really well tonight. So, we stayed in the game, got the dominance back step by step and then Shaq closed the game. Unbelievable, it's really nice and I think absolutely deserved tonight.
"You need a bit of luck, what we needed for both goals with the deflections, but bringing the situations, having these moments, that was football, huh?
"The first goal: what a goal, what a goal. It was brilliant and exactly like they have to do it – fantastic pass, super control and a fantastic finish, that's how it should be. It was a nearly perfect night for us."
Liverpool are now 19 points ahead of Jose Mourinho's United after just 17 games, remaining unbeaten in the league all season.