The 18-year-old Belgian winger, who had earlier been booked for diving, dragged a misfiring United back into the game with a calmly-taken 55th-minute equaliser and then smashed a dipping volley past keeper Keiren Westwood to win the match just six minutes later.
On an afternoon when United were far from their best, they were all but taken apart by the bottom-of-the-table Black Cats before the break with Craig Gardner's fifth-minute opener setting the stage for a concerted assault by the home side.
And Carrick, who played alongside Januzaj, believes he is good enough to "be anything he wants to be".
"He's done it on the big stage and he's made a big improvement and 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," Carrick told Sky Sports 1.
"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."
Carrick also denied that United let their heads drop after they went a goal down.
"We never really believed we were in a bad position," he said.
"We controlled the game and gave them a couple of chances early on. Our backs were against the wall but we knew we had a chance to win the game. Credit to the boys for believing and we stuck together.
"The character and the desire is till there. As a team we have good and bad days and we've had some bad days over the last few weeks. We relished the challenge of the game."
Manchester United manager Moyes was also impressed with the young midfielder's double goalscoring debut and revealed he came close to giving him his first start a few weeks ago.
Moyes said: "I remember giving Wayne (Rooney) his debut and Adnan is certainly that quality. He's a top, top player.
"I would have played him two or three weeks ago but it didn't work. We just never quite got the opportunity for different reasons and I thought today was the right time and it was hard to stop him.
"He's going to be a special talent and he showed he's got goals and ability. We will keep his feet on the ground."
Moyes is not worried about keeping Januzaj at Old Trafford after his two goals against the Black Cats will have made other managers sit up and take notice of him.
"Every young boy wants to play for Manchester United and as a young player I can't see anywhere else a young boy would want to go."
Sunderland caretaker manager Kevin Ball was frustrated after a one-goal lead against the defending champions turned into a 2-1 defeat.
When asked what his feelings were after the game, Ball said: "Probably a bit of disappointment having done so well in the first half and scored the goal.
"We knew it was going to be difficult in the second half. The players and myself are disappointed as they worked very hard today."