"We had high expectations of wanting to play well and to win but it didn't work out for us on the night," Carrick told MUTV.
"We started very well and made chances and if we'd scored one then maybe it would have been a different game. After that, things didn't go right for us really. We still had opportunities to score but we just didn't take them.
"Credit to Barcelona though - they're a good team with good players and they deserved to win tonight. We can't argue about that.
"We had gone unbeaten in this competition for two years and sadly it's come to an end on the wrong night. I think we can be proud of what we've achieved in the last couple of years, but it's hard to take.
"However, the club moves on, we have to move on and come back stronger next season. We've got to defend the Premier League title and go again in this competition.
"That's what we're here for."