The Red Devils boss was unhappy with the way his side surrendered its lead at the Allianz Arena on Wednesday evening
United took the lead in the quarterfinal second leg at the Allianz Arena 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.
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 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 equalize 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 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."