Robin van Persie's 21st goal of the season put United ahead in the 19th minute and Nemanja Vidic doubled the lead just after the break - inadvertently deflecting in Patrice Evra's header.
Substitute Daniel Sturridge pulled one back, his second in as many games after joining from Chelsea for £12million, when he fired home after goalkeeper David de Gea had parried Steven Gerrard's shot, setting up a tense finish for the Barclays Premier League leaders, but they ground out victory.
"I felt we were too tentative in the first half - but you can understand with young players playing here," he said.
"There wasn't much in the first half but any chances they had were through us giving the ball away.
"I said before we went out that when you play a rival - and this rivalry is massive - you need courage and you need to have belief.
"We wanted to pass the ball, which is great, but it has to be aggressive and we presented them with the ball and three opportunities just by being tentative.
"I am very proud of the fight of the team in the second half having gone 2-0 down. Teams who come here often crumble but I thought by the end we deserved something to be honest.
"We know in terms of the points we are 24 behind (United) but we are not 24 points behind in quality. The points difference is due to the squad and that is the difference for us, the depth of the squad.
"Once we close the gap in the squad over the coming transfer windows I have great faith we will be able to challenge and I have seen enough hope in these opening months that we will do that."