The Red Devils, who had made their worst start to a season in Premier League history, looked to be under severe pressure when they fell behind to Craig Gardner's opener after just four minutes.
However, the English champions hit back in the second half with two goals from 18-year-old winger Adnan Januzaj.
"We never really believed we were in a bad position," Carrick told Sky Sports News.
On Januzaj's performance, Carrick said: "He's done it on the big stage and he's made a big improvement. He's taken his chance in pre-season and he took it on his first start, two great finishes. He's won us the game.
"He's got a great attitude. He can be anything he wants to be, he's that good. I'm sure he's not going to get carried away with himself. The second [goal] was a terrific volley."
United's victory at the Stadium of Light propels them to ninth place in the Premier League table.