United took the lead in their quarter-final second leg at the Allianz Arena on Wednesday through Patrice Evra's thunderbolt strike, a goal that gave Moyes' men the lead on aggregate and cancelled out Bayern's away goal from the first leg.
But Bayern responded within 22 seconds of the restart to level through Mario Mandzukic, the Croatian getting ahead of Evra to head home Franck Ribery's cross.
That was the impetus that 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 said.
"The 30 seconds after we scored the first goal is 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."
Defeat ends any hopes United had of lifting a trophy this season, and with their inconsistent form in the Premier League they may not to make the competition next term.
"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."
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