Liverpool scored with their first real attempt of the game, as Virgil van Dijk went unchallenged to power in a header from a corner.
The Reds had two other goals chalked off in a first half they dominated.
And the pattern of play was similar in the second period as United’s goal was under siege.
The visitors grew into the game slowly and could have equalised as Anthony Martial blazed over after some fine build-up play from the Red Devils.
But Mo Salah sealed the win for Liverpool in the last minute as Alisson found the striker brilliantly, allowing the Egyptian to run through and slide the ball under David de Gea.
Klopp told Sky Sports: “It’s a big relief, I was really happy with 85-90% of the game, we were brilliant. We dominated the game, especially in the first half. The energy they put on the pitch was incredible.
“On a normal day we would have scored three times in the first half and in the second half until 65 minutes we should have been more clear.
“But then United have obvious quality, played a bit more football and we had to defend. There we little mistakes here and there, we didn’t use possession well enough and so the game stays open.
“Then we scored a wonderful, wonderful goal at the end, a really good feeling.”
Klopp was asked if he was surprised by the level of performance his players maintain.
He added: “To be honest, not really. We had exceptional sessions against the kids – 11 v 11. The boys go like crazy. I don’t take it for granted but I see them during the week so I’m not surprised.”
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