United took the lead in their quarter-final second leg at the Allianz Arena on Wednesday through Patrice Evra's thunderbolt, a goal that gave Moyes' men the lead on aggregate and cancelled out Bayern's away goal from the first leg.
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That was the impetus the defending champions needed and Pep Guardiola's side went on to secure a 3-1 win on the night, 4-2 on aggregate, as Arjen Robben and Thomas Muller added the second and third.
And while Moyes was proud of his team's performance, he conceded that allowing Bayern to equalise so quickly was a costly error.
"I am pleased with their (the players') performance, but a couple of small errors cost us in the end," he told reporters.
"The 30 seconds after we scored the first goal were poor, as a schoolboy you are told not to concede a goal straight away after scoring.
"We conceded too quickly. It's always going to be tough, but after the first goal you need to give five or 10 minutes (without conceding).
"I don't know if the concentration went after the goal, but we had enough experience on the pitch to do that.
"I never thought we were out of the game, we needed to score one more goal but obviously the third one killed us."
Moyes, however, is confident United will soon return to Europe's premier club competition, despite having little chance of qualifying through the Premier League this season.
"We've not got (the) Champions League (next season) but I believe it is not far away," he added. "Hopefully it will only be one year (away) where we have to rebuild.
"My focus now is on getting this team back in the competition.
"There is no shame in losing to Bayern Munich in the quarter-finals. We played really well and showed the qualities we've got."