The 170th league meeting between the two sides was decided by a Marcus Rashford brace, who scored twice in 10 first-half minutes to move the Red Devils five points clear of their opponents in the table.
Although the Reds played admirably in the second half and halved the deficit when Sadio Mane's cross was turned into his own goal by Eric Bailly, they could not go on to find an equaliser.
"In the second half we did well but the first half of course was not really good so it's frustrating in the end," goalkeeper Karius told the official Liverpool website. "Because you ask yourself why we didn't play as good as we did in the second half after the game.
"That's our own fault, we have to ask ourselves that question. It's not easy to get something here if you only play good for one half so, of course, it's disappointing. At the end, we could still get a point but the first half wasn't really good from us. We didn't defend the counter-attack so good when they had their moments.
"They didn't have a lot of moments but when they did they were really dangerous so we could have done better as a team. It's difficult to say after a game what exactly what it was but as a team we could have done better and defended the situations better."
Liverpool will look to bounce back from the defeat to their arch-rivals when they host Watford at Anfield next Saturday.