Rooney was reacting to Sunday's win at Arsenal, which came a week after Carrick's winning debut against Manchester City.
The former United captain's son, Kai Rooney, made his FA Youth Cup debut last Friday in victory over Derby.
"The change in Manchester United over a two-game period is incredible," Rooney said on the latest episode of The Wayne Rooney Show.
"Michael and his coaches deserve a lot of praise for that.
"Just putting an attitude into the players and a focus, and a togetherness into them which, if we're honest, we didn't see under Ruben Amorim.
"Manchester United went there believing they could win the game. They didn't create a whole lot of chances, but you always felt they had a good control of the game.
"Arsenal started well. And then when Manchester United went a goal down, the reaction was great.
"Manchester United deserved the win."
Carrick calm
Rooney also revealed: "There's a calmness about the place.
"I was watching the under-16s on Saturday morning and they were all there again. Michael Carrick was there, Steve Holland, Jason Wilcox, all the coaches were there and so on, which you've heard me talking about a lot, that connection with the academy.
"What I saw, not just from Michael, but the other coaches, Johnny Evans as well, is a real calmness and a belief.
"Michael told me how good the players were and he felt he had to stop the session early because they were that good and he wanted to bottle it up.
"And that's proven today. An absolute incredible performance."