Two late goals from substitute Xherdan Shaqiri sealed the points for Liverpool, but Klopp was delighted with the performance up to that point in a game that his side dominated.
"It was a brilliant start, one of the best performances we've had since I was at Liverpool, to be honest. The first half an hour was outstanding. How we played was just perfect," Klopp said.
"I'm completely over the moon about the performance. The way the boys played tonight was just outstanding. We saw the line-up of United and their quality - how well organised they were and they fight from the back, you knew it could be really tough.
"But the plan was to take the risks, be brave and chip the ball behind the lines. The first goal was how, if you could have drawn, it would have been the goal. It was just brilliant. A fantastic performance, it's so deserved. Top, top first-class team.
"I don't think United wanted to play defensively but we didn't let them out. Overall it was just brilliant."