United lost the title on goal difference to rivals Manchester City last season and the midfielder has urged his side to kill games off more like they have done in previous campaigns.
Carrick told reporters: "It has been a while [since we were in such a habit]. We had a spell, about a year or two ago, when we were starting games really well and killing teams off really early.
"We want to get back to that, because Old Trafford is a difficult place to play when you go a goal down. We want to make it that way, certainly in the games that are to come because going a goal down has made things a lot harder for us than they should have been.
"Keeping a clean sheet gives us something to build on in the future.
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"They had a couple of opportunities and, when they put balls into the box like they did, those can fall to anybody. We got the clean sheet. It is something to build on because the performance needs to be better.
"A second goal would've put us in a much better position going into the final few minutes but, in the end, we held on and we're still in a good position at the top of the league."
Team-mate Jonny Evans agreed: "We didn't play well. The game was a bit slow and laboured. It was fortunate that we got the goal early doors but we didn't really kick on from that.
"I was thinking at the time: 'This is great, we've got a goal early doors and we can score a lot here', because everybody is looking at the goal difference, especially after what happened last season.
"I think we all know we didn't play to our usual standards. We'd had a tough game at the weekend [against QPR] or at least we made it tough for ourselves.
"It was quite a physical game, a lot of running was done and it was probably one of our highest running stats of the season so there were some tired legs in the squad and I think it was all about seeing it out to get the three points.
"The clean sheet is good for the confidence, even if we didn't play in the normal United way. We probably were a bit more cautious going forward and holding our positions a bit more."