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Ashley Barnes scored the only goal of the game with seven minutes to play, converting a penalty he won himself after colliding with goalkeeper Alisson.
It was Liverpool's first Premier League defeat since April 23, 2017, when their former striker Christian Benteke got both goals in a 2-1 win for Crystal Palace.
The result leaves Liverpool six points behind leaders United at the top of the table at the halfway stage of the season. City are four points ahead in second, but also have a game in hand.
Klopp raised eyebrows with his team selection at kick-off, with Sadio Mane the only member of his usual front three to start in attack. Divock Origi, Xherdan Shaqiri and Alex Oxlade-Chamberlain started, with captain Jordan Henderson missing through injury.
Since the start of last season, Liverpool have now played 12 games without Henderson in the league, losing four. They have lost only two of the 45 games in which he has played.
Origi had the best chance of the game for Liverpool. Put clean through on goal, he looked to pick out the top-right corner with a curling effort but saw his shot cannon back off the bar and away, with Mane free to his left for a simple finish.