United striker Robin van Persie scored the crucial goal within 60 seconds of the kick-off, but Andy Carroll and Carlton Cole both came close to netting an equaliser in the second half.
It was only the Red Devils’ third clean sheet in the Premier League this term, and Carrick hopes the 19-time English champions can take confidence from an improved defensive display.
“It was by far not a classic,” Carrick told United’s website.
“Obviously we weren’t at our best and the flowing football that you want to play and the chances that you want to create didn’t quite happen. But we got a clean sheet which was quite pleasing and I thought we defended well.
He added: “A clean sheet is the biggest plus that we can take from tonight. They pumped a lot of balls into the box, asked questions and made us defend. Our two centre halves were terrific. West Ham had two big guys – as Carroll goes off, Carlton Cole comes on – and it can be tough.
“But 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. We are quite satisfied.”