Marcus Rashford scored the only goal of the game as United secured a win that takes them to within a point of City in the table and puts them firmly back in the race for the top four.
Louis van Gaal's side have faced a difficult week, having only just salvaged an FA Cup replay against West Ham before being knocked out of the Europa League by fierce rivals Liverpool on Thursday.
And Carrick admits that winning at the Etihad Stadium could be pivotal in helping United salvage something from a disappointing campaign so far.
"It was a good first half, a great goal from Marcus, an unbelievable finish," he told Sky Sports. "It has been a tough week. there were bodies cramping out there.
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"It was a great result and a good way to bounce back from Thursday night. It is more than three points, as it was a must-win today.
"It gives us a real boost, something to go at and momentum for the final stages of the season."
Morgan Schneiderlin was keen to heap praise on Rashford's all-round contribution and insists his side fully deserved their victory.
"It's amazing for him," he said. "It's not just because he scored, it's his whole game, the way he keeps the ball up front. He's a great prospect for Man United. We're very happy for him.
"We deserved the win. In the first half we didn't let them play, we did some good pressing and we had a few chances.
"The legs were a bit tired but we showed good character and mental strength. We're very happy to win, especially for the fans, and we're closing he gap to third and fourth.
"It can change from one week to another but we are confident we can do it. A club like United must be in the Champions League."