United twice led after goals from Bryan Mbeumo and Benjamin Sesko, but conceded in the 96th minute as Ainsley Maitland-Niles scored with a scorching drive from 35 yards to snatch a point.
Everton’s first goal from Tyrique George was also a fine strike and Carrick was left to rue his side’s misfortune at the Hill Dickinson Stadium.
"They're two fantastic strikes from in and outside the box, so they don't go in too often. I thought we did a lot of good things in the game to get us in the position we were in twice,” he told the BBC.
"Some of our attacking play was really good and we looked dangerous and did a lot of things that we were trying to do. And for the most part, I thought we dealt with them quite well.
"It hurts as we came here to win and we put ourselves in a good position."
Carrick withdrew Marcus Rashford on 70 minutes after a lively performance on the left wing.
"He felt a little bit of something, so we had to change him," Carrick said. "We had good connections pretty much all over the pitch at different times of the game, and Marcus and Luke Shaw have played together for a long period of time and have a good understanding.
"And you can see that, they look really dangerous down that side, and there's obviously more to come."
United next host Sabah from Azerbaijan in the Champions League on Thursday.
English Premier League
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