Cunha played his part in United's win at Everton with a crossfield ball to Bryan Mbeumo, who then laid on the pass for Benjamin Sesko to score the only goal. Cunha has six goals and two assists for United since completing a £62.5m move from Wolves in the summer.
Carrick praised the Brazilian for his link-up play with Mbeumo and Sesko and highlighted his defensive work as United dug in their heels to win their first visit to the Hill Dickinson Stadium.
"He had to defend in the second half, Matheus, and he did it really well," the United boss said. "It was some pass over the top. We needed a whole team performance in terms of the pressure Everton put us under, and the boys did it. If you do that, you get rewards because we’ve got dangerous players."
As well as helping United finish in the top four to qualify for the Champions League, Cunha has his sights set on the World Cup with Brazil. He was called up for the last squad in November and is eagerly awaiting another call for the friendlies in March against France and Croatia.
"Of course the main focus is helping Manchester United, but I cannot hide the motivation of defending the Brazilian national team, especially at a World Cup," Cunha added. "I'll do everything to be in that group, to represent that shirt. It is every player’s dream. The competition is strong, everyone knows the quality of Brazil’s forwards, but I trust in myself and in the work I'm doing. I am living a good moment here at United and I will keep working at the maximum to maintain this level of form and even improve it.”