Captain Nemanja Vidic gave United the lead at Old Trafford on Tuesday in the first leg of a quarter-final tie against the European champions, before Bastian Schweinsteiger grabbed a crucial away goal for Bayern.
While the Germany international was later shown a second yellow card for a foul on Wayne Rooney, his strike handed Bayern a slight advantage ahead of next week's second leg in Munich.
Despite missing the opportunity to inflict a rare defeat on Pep Guardiola's men following Vidic's opener, Moyes feels his side have a strong chance of progressing to the last four.
"We know we have to score a goal (in the second leg)," he said.
"If we don't score, Bayern will have done enough to go through. We've given ourselves a great opportunity to try and get a win.
"We said before we want to give ourselves a fighting chance in Munich. We've done that.
"The players played really well. I'm delighted with the performance, but disappointed we conceded a goal.
"It was a gutsy performance, the crowd were great, it was a great atmosphere."
