After a goalless first half in which the home side had the better of the chances, Wayne Rooney scored the vital winner in the 78th minute when he rifled home after Marouane Fellaini’s effort had hit the crossbar.
Liverpool had a number of chances to take the lead before half-time, with Adam Lallana guilty of spurning the Reds’ best opportunity in the first half.
And Carragher admitted after the final whistle that Liverpool’s lack of goal-scoring threat was always going to be a problem for Jurgen Klopp’s men.
Speaking on Sky Sports after the game, Carragher said: “Before the game you were thinking, who will get the goals? It’s been a problem for Liverpool all season.
“I said at half-time that the chance Lallana had probably summed up his Liverpool career. He hasn’t scored since May – he’s a £25m signing, an attacking midfield player… it has been a major problem.
“We said at half-time how poor the game was. I think Liverpool were the better team but it doesn’t matter in games like that – maybe [it will matter] in the bigger picture.
“But today, for the United fans and players, these games are about winning – and that’s what they’ve done.”
The result lifted Manchester United up to fifth place in the Premier League table and just two points behind fourth-placed Tottenham Hotspur.
Liverpool, meanwhile, are ninth and eight points behind Spurs.
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