Christian Benteke was Liverpool’s match-winner for the second time in five days as he plunged Sunderland deeper into relegation trouble.
The Belgium international, who scored the only goal against Leicester on Boxing Day, did it again at the Stadium of Light when he took advantage of some flat-footed defending 22 seconds into the second half to clinch a 1-0 victory.
Liverpool boss Jurgen Klopp was delighted his team found a way to win.
“We were the better team and you could see that the pressure on both teams was big,” he told Sky Sports 1.
“A little bit more for Sunderland because of the intensity of the last weeks and months. You could help yourself if you got an early goal, we didn’t but we did not do too bad in situations against a very good defending team.
“They were only defending so that makes it pretty hard for us because you don’t get space and you always have to go again and slow down the game.
“Then we made the goal after half-time and we controlled something like the game.
“They were waiting for set-plays, they are difficult to defend but we did it. We could have got made a second or third one, we didn’t, we had to wait until the final whistle.”
Allardyce felt the Black Cats were unfortunate in the manner that Liverpool got their goal but praised his side’s effort, although he conceded a lack of killer instinct in the final third.
“We didn’t really produce enough quality,” he said.
“When we did we got good chances, we couldn’t finish them off but we could have done more with the opportunities in terms of our final third quality - we need to try and get that a little better.
“A resilient performance, one where Liverpool never really looked like creating too many clear-cut chances or opportunities and if we can build on that scenario and get a bit more luck then we’ll try to go up the other end, get a result and score a goal.
“The injuries are a problem. We’re facing another centre-half who’s come off injured so we’ll have to wait and see how Seb (Coates) is.
“I thought Wes (Brown) did great today. I thought the team did well overall, a good shape to the side. They’re playing a Liverpool side that’s got a huge amount of money spent on it and often matched them.”