The Reds were nearing a first-ever win for an English side in the Donbass Arena after Danny Welbeck s goal earned United the lead, but Taison equalised 11 minutes from time to ensure the spoils were shared.
Carrick's performance against Shakhtar earned him the nod as Man of the Match from the club's official Twitter account @ManUtd, but the midfielder was frustrated by the final result.
"It is a good point, but it is slightly disappointing because we were in front," he told ManUtd.com. "We weren't really in trouble, and we were controlling the game through our defence. To lose a goal is disappointing, but a point is not bad."
The Ukrainian side had never previously dropped points in their three previous Champions League meetings with English teams on their own patch, but Carrick maintains that the Reds' expectations mean that the draw took some of the gloss off a solid performance.
"We expect to do better because we have high standards. We hoped to go there and win, and we are disappointed that we haven't. Maybe some teams, after the run we have had, would look at this point and be delighted. We are a tad frustrated that we haven't come away with anything but it is not the end of the world."
With United on the road at Sunderland on Saturday for a 17:30 BST kick-off, Carrick is calling upon the side to build upon Wednesday's encouraging European display.
"I think we just get back to doing the right things. If we do that, in time, we will be fine. You have periods, whether when things don't quite go right. You just have to find a way to get through it or face it, you can't hide. You have to face it and respond, and