Having rarely produced a clinical performance all season, the Reds delivered one when they needed it most to keep their top-four hopes alive.
Klopp's side are now four points behind fourth-placed Chelsea with a game in hand.
Speaking to BBC Sport, the German said: "Top. It was exactly what we needed in this situation. We are in a situation where we are under some pressure and need to win the games to have a chance. We came here to play a team in form.
"It was tough and the start was not good. We struggled to defend on the wings. They outnumbered us there and it took us a while to adapt to that.
"When we started doing that we were really in the game and scored two goals. Everybody knows this season that the second half is usually United's half. They usually turn games around but we denied that.
"We played a really good first part of the second half, scored a third goal and then it's an open game. It's intense with a team chasing the game and one team defending. The fourth goal was brilliant. Top again from Curtis [Jones].
"We've not had this result before so we must have done something right tonight."