The small pocket of away fans inside the Allianz Arena started believing their team could pull off another shock when Patrice Evra scored in the second half to give United a 2-1 aggregate lead against the Germans.
But United's defence then switched off, allowing Mario Mandzukic to score just 22 seconds after the restart.
That goal triggered a United collapse. Thomas Muller and Arjen Robben found the net to seal a 4-2 aggregate win for Bayern which dumped Moyes' men out of the competition at the quarter-final stage.
"It's the sort of thing you learn as a schoolboy once you score a goal, make sure you don't concede," the United manager said.
"You get into position and do your job. We had concentrated so fabulously well throughout the game.
"The players had done tactically well. Their positioning was good and they did their jobs.
"For us just to give it up on that moment, the goal up, was really disappointing."
Bayern dominated most of the play, ending the match with 63 per cent possession, but they could not find a way past David de Gea in the first half despite 13 attempts at goal.
Indeed it was United who had the better chances in the first half.
Wayne Rooney was the guilty party as he raced onto Darren Fletcher's pass, only to dally on the ball in the box before fluffing his shot.
The England striker, who played the game after having a pain-killing injection in his toe, then missed an even easier chance in the second period from eight yards.
Moyes insisted before the match that he was not gambling with Rooney's long-term fitness by fielding the player, but after the match he admitted the striker's performance was not up to scratch.
"I thought at times he was having a struggle striking the ball on some occasions," the Scot said. "I was concerned that he hadn't trained all week and he had taken an injection.
"But he is so vital a player to us, especially with Robin van Persie out (injured). But he just had a couple of opportunities and he couldn't quite take them."
Although United did well to keep pace with Bayern in the first half, the gulf in class between the two teams was clear.
Bayern were without Bastian Schweinsteiger and Javi Martinez, but they were still able to field a stellar midfield that contained Toni Kroos, Mario Gotze and Muller, with Ribery and Robben providing the ammunition on the flanks.
With United seventh in the Premier League, this was Moyes' last chance to get his team back into Europe's elite club competition, so next year will be the first time in almost two decades that there will be no Champions League football at Old Trafford.
But Moyes insists the future remains bright for the club despite their woes.
He said: "We've not got Champions League football but I believe it's not far away and hopefully it's only one year.
"We've got to rebuild and our focus now is getting a team together to get back in this competition because it's a really good competition and we've really enjoyed it."
Moyes insisted he had never been told that Champions League qualification was a minimum requirement, and he said he would have no trouble attracting players to Old Trafford despite a lack of top-class European football.
"That's not a question for me," he said when asked about his position. "My job is just to get on with my job and my focus is on getting a side together to get back in the Champions League. That's our main objective now.
"The club have never had any problems and everything we've talked about, we're looking to spend the right money on players who are available and it's not anything to do with Champions League.
"Any players we've quietly discussed it with are more than happy to join Manchester United. They know it's not a long-term thing."
Pep Guardiola, of course, has far fewer problems than his opposite number.
Bayern have already won the Bundesliga, they are in the semi-finals of the domestic cup and now they have a last-four meeting in the Champions League to look forward to.
The Spaniard conceded that United made it hard work for his team on Wednesday night, though, due to their defensive attitude.
"I knew it would be hard," he said.
"It's not easy to play against a team when they have eight players in their box. It happened in Manchester too.
"We decided to create a bit more passing in midfield and get the ball wide. They defended well, but when it got to 1-1 we played much better and we deserve to be in the semi-finals."
Bayern will find out whether they draw Chelsea, Atletico Madrid or Real Madrid on Friday.
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