Casemiro opened the scoring with a header in the 11th minute before Benjamin Sesko doubled the lead, with Bruno Fernandes providing his 19th Premier League assist of the season in the build-up. Mathias Jensen pulled one back late on but United held firm.
Fernandes is now two assists short of the Premier League all-time record with four games remaining.
Carrick spoke to MUTV and Sky Sports after the final whistle.
"I thought the first 20-25 minutes was as good as we've started a game — really creative, dynamic with quick football. Positive, good connections through the middle and we found Bruno quite a lot behind their line. We probably could have had an extra goal or two at that stage."
On managing the second half: "We tweaked things second half and, for the most part, had decent control. I thought the spirit and effort was fantastic. We could've scored more but, at this time of year, it's a results business. There were throws coming in from all angles, we had to defend and everyone played their part today."
On Fernandes's record chase: "We hope so, of course. Bruno is a big part of it but it's the forward line collective, their relationships. I thought we looked dangerous enough throughout the game."