David Moyes's men were seen as underdogs against the defending champions but produced a valiant display to keep the tie in the balance, though Bayern hold a potentially crucial away goal after Bastian Schweinsteiger cancelled out Nemanja Vidic's opener.
The German midfielder was then sent off in the 90th minute for a second bookable offence following a foul on United striker Wayne Rooney and, despite the holders' away goal, Carrick is hopeful that United can seal progression in the return leg at the Allianz Arena on April 9.
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"We are still in the tie and we have a chance going over there.
"It was a bit frustrating going in front and then conceding a bit of a soft goal but, overall, I think it was a pretty good performance.
"They are a good team and had the ball for long spells but I thought we defended well as a team and, on the counterattack, at times we looked dangerous.
"We knew it was going to be a tough night - it was never going to be easy - so to get the draw is probably a fair result.
"We are not getting carried away, we know the task ahead will be tough. We are going to have to be right at our best to get the right result over there.
"We still have a chance, we have to believe so we can go over there and be really positive."