Carrick’s side thought they had earned a hard fought 2-1 win when substitute Benjamin Sesko restored their lead in the 88th minute.
Everton were down but not out, however, with new signing Ainsley Maitland-Niles scored a sensation long-range equaliser in the 96th minute.
Man United now have just four points from a possible nine, and sit down in 11th in the Premier League ahead of the Manchester derby in their next fixture.
Speaking to reporters after the game, Carrick admitted the late draw with David Moyes’ side was disappointing.
“There was a lot of good things in the game for us. We played some really good football, were dangerous at times,” he said.
“For the most part we defended well as a team. The longer the game went on there were a few more balls into the box and a few decisions we made it more difficult for us when we should’ve seen it out really. We’re disappointed we didn’t come away with a win.”
What mistakes were those? “I’ll keep them to myself a little bit but we should’ve seen the game out really so we’re disappointed with that.”
On the Everton goals: “There were two great strikes to be fair. We didn’t give them real clear chances. We defended well. They had a few bits that dropped around the edge of the box.
“We defended the box pretty well for the most part. We look to the reasons how they ended up getting those opportunities but those were really good strikes.”
On the game in general: “There was a lot of good things. Especialy early on in the game and probably towards the end not so much.
“We looked dangerous. I thought we called a lot of problems. They got really good goals for different reasons and our second one should’ve been a winner.”