The 31-year-old played a poor backpass to David de Gea, allowing Jay Rodriguez to open the scoring before the visitors were eventually beaten 2-1 by the Red Devils
Sir Alex Ferguson's side carved out a seven-point lead at the top of the Premier League thanks to a hard-fought victory against Mauricio Pochettino's side at Old Trafford.
But a shock had appeared to be on the cards in the third minute as Jay Rodriguez latched onto Carrick's weak backpass before sliding the ball home - and the 31-year-old accepts he was to blame for the goal.
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"As it popped up I just tried to flick it back to David and I didn’t get enough on it. But we bounced back from that and got the goal and played some decent stuff in the first half.
"To be honest, I thought we defended well, it was my mistake that led to their goal and you can’t really legislate for that, it wasn’t really a defensive error in terms of the team."
Carrick also criticised his side's second-half performance, but was keen to praise his side for seeing the result out.
He added: "We showed character, it wasn’t an easy game and we are happy with the three points.
"I think I gave us a bit of a bad start but we bounced back in the first half and played some good football.
"For the majority of the first half, we were quite happy with our performance but the second half was a bit of a write-off to be honest, we didn’t play well at all.
"We just found a way to get over the line so we are delighted to get the three points. There are a few things to work on but we can still take the positives and we are still in a good position.
"We are delighted with the three points, we knew it was a big game and it was a terrific result for us."