The 35-year-old hadn't featured under Mourinho this season before he made a goalscoring appearance in the 3-1 EFL Cup win over Northampton Town on Wednesday night.
"The way we started the game, we could have made it much easier for ourselves," Carrick told Sky Sports.
"We took our foot off just before half-time but then we finished it off.
"I don't have to prove myself.
"I've got my own game, I'm not going to change my game. The manager knows what I can do."
The veteran curled home a lovely first-time effort in the 17th minute as United progressed to the fourth round despite a scare just before half-time when the Cobblers equalised.