Wayne Rooney had a penalty saved before the returning Paul Scholes opened the scoring on the stroke of half time.
Second-half goals from Danny Welbeck and Carrick helped secure the win for the Red Devils.
"I thought we played some good stuff at times in the first half and we created opportunities that we could have put away and we started thinking 'is this going to be one of them?' but the goal right before half time gave us momentum going into the second half," Carrick told BBC Sport.
"The second goal then put us in a great position and then it was nice to cap it off.
"It's always great to score. I managed to get one the other week at QPR. This one was slightly different but it was great to see it go in."
United will now turn their attentions to next Sunday's league match at Arsenal.