Benjamin Sesko struck on 71 minutes to earn the points from the Toffees.
Carrick said afterwards: "I think there is games where you have to do that and show different sides to the game. We had the first four games, four wins in a row and some really good performances in there, so I expected a period where we'd have to find ourselves a little bit and find results.
"I thought we did it at West Ham and we did it today.
"That spirit, that sacrifice and that willingness to work for each other and as a team the way we had to defend the box for the last period of the game - everybody playing their part - and in the end a clean sheet was fantastic. A great win for us."
On his side's performance, Carrick also said: "It didn't really come easy for us tonight. I've come to Everton for a long time and every time we come over here it's tough. I've never had an easy game. I've never had a game where it's all clicked. We expected a tough one tonight.
"When we played our last game, it is what it is. The lads have trained really well and we came in looking nd feeling good bout the game. It was one of those nights for us. We had to dig deep and it was a bit of a struggle at times. But sometimes they are the best ones to win."