United twice took the lead through Bryan Mbeumo and Benjamin Sesko, but Everton refused to give up and Ainsley Maitland-Niles produced a stunning 35-yard strike in the sixth minute of added time to rescue a point.
Everton had also scored through an impressive effort from Tyrique George, leaving Carrick frustrated after his side had twice been in a winning position.
"They're two fantastic strikes from in and outside the box, so they don't go in too often," Carrick told the BBC.
"I thought we did a lot of good things in the game to get us in the position we were in twice.
"Some of our attacking play was really good and we looked dangerous and did a lot of things that we were trying to do. 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 was also forced to replace Marcus Rashford in the second half after the forward impressed on the left wing.
"He felt a little bit of something, so we had to change him," Carrick said.
The United boss was pleased with the understanding between Rashford and Luke Shaw, who combined effectively down the left before Rashford's withdrawal.
"You can see that, they look really dangerous down that side, and there's obviously more to come," Carrick added.
United will now turn their attention to the Champions League when they host Azerbaijan side Sabah on Thursday.