Casemiro and Benjamin Sesko struck the goals for United on the night.
Carrick said afterwards: "It was quite an open game at times. We started fantastically well. Credit to them, they are a good team and we knew they were a good team.
"They came into the game a little bit and were in a good place at half-time so we tweaked things second half. Towards the end they had a few opportunities but I thought the spirit, effort and togetherness of the players was fantastic. It's the time of year when it's a result business and we needed to take points."
On another assist for Bruno Fernandes: "We have scored nearly every game and Bruno is a big part of that. He could have maybe had one more and scored himself."
Asked about seeing the game out after Brentford pulled a goal back, Carrick concluded: "It's a tough finish in a tough league. To get the amount of wins we have, we are delighted with that.
"Today was all about the three points for us. It's another win. We have an important game coming up and we can look forward to that."