United crashed out of Europe on Wednesday after losing 3-1 to defending champions Bayern Munich in the return leg of their quarter-final, which completed a 4-2 aggregate defeat.
David Moyes' men went ahead on aggregate just shy of the hour mark courtesy of Patrice Evra's stunning left-footed effort.
But United's joy was short lived as Bayern hit back through Mario Mandzukic within 22 seconds. The Bundesliga champions duly added a further two goals to settle the tie.
"It's bitterly disappointing, we put ourselves in a great position after scoring the first goal," said Carrick.
"It was a big blow to concede again after 22 seconds.
"Wayne (Rooney) had a great chance to score again later on and it didn't go for us.
"The high of scoring and conceding is a blow but it is something we had to deal with and we were still well in the game, the second goal was a bit of a killer."
United are seventh in the Premier League, seven points adrift of fourth-placed Arsenal, and look set to miss out on a spot in Europe's most prestigious club competition for the first time since 1995-96.
