The latest substandard display of United's increasingly troubled season saw them throw away the lead to go down 2-1 in the last-32, first-leg contest in Denmark - a result set to be looked back upon as one of the most embarrassing in their European history.
Eight hundred Red Devils supporters made the trip to the MCH Arena in Herning for a match that cost £71 per ticket to attend, and it was clear what some of the fans thought of what they had seen for their money at the end, with chants of "we're f*****g s**t" coming from the away section.
United boss Louis van Gaal subsequently admitted when asked about the criticism: "Maybe they are right."
And midfielder Carrick, who wore the captain's armband in the absence of the injured Wayne Rooney, told MUTV: "I can't knock the supporters, they were incredible.
"They've been great throughout my time at the club, especially during the last couple of years when they haven't had much to shout about.
"They've stuck with us and travelled to support us so I can only apologise.
"Hopefully we can give them some reward for the unbelievable support they've shown both the players and the club."
United had 13 senior players they could not call upon for Thursday's game due to injury and illness, with goalkeeper David de Gea the latest to join the list after sustaining a knee problem in the pre-match warm-up that forced him out of the first XI.
But Carrick stressed: "You can talk about injuries but, irrespective of that, we can't make any excuses.
"To come here and lose is nowhere near good enough.
"The only positive is that we've got another chance at it (in the second leg), but it doesn't really change the fact that we were simply nowhere near good enough."
Source: PA