An uninspiring quarter-final, second leg burst into life when Evra met a bouncing ball with a rasping half-volley to put United ahead in the 57th minute and 2-1 in front on aggregate.
Mario Mandzukic equalised also immediately for Bayern with a header and Thomas Mueller turned in an Arjen Robben cross in the 68th minute before Dutchman Robben added the third himself.
Wayne Rooney had missed a chance to put United 2-1 ahead on the night when he scuffed a shot from in front of goal and Bayern went though 4-2 on aggregate to stay on course for a treble and continue United's miserable season.
Atletico stun Barca to book a place in CL semis
Atletico Madrid hit new heights in their incredible season when Koke's goal secured a 2-1 aggregate success against Barcelona on Wednesday that put Atletico into the last four of Europe's elite club competition for the first time in 40 years.
The stunning 1-0 victory at an electric Calderon stadium followed last week's 1-1 draw in Barcelona in their Champions League quarter-final, first leg and denied Barca a record-extending seventh consecutive appearance in the last four.
A fired-up Atletico blew Barca away in the opening 20 minutes and after midfielder Koke volleyed in from close range in the fifth minute former Barca forward David Villa twice hit the crossbar.
"It was a very important goal because it got us through to the semi-finals," Koke told Spanish broadcaster Canal Plus.
"The work of the team across both legs has been spectacular
"We ran a huge amount, we played the two games in the way we wanted and I think we were the clear winners," he added.
"With these fans behind us it is very hard for us to lose a match here."
Barca showed more intensity and had their chances in the second half against an exhausted Atletico but the home side held out with inspirational coach Diego Simeone willing them on from his technical area.
"We had the game within our grasp," Barca midfielder Xavi said. "We had four or five very clear chances that normally we would put away.
"They made the most of those first 15 to 20 minutes when they played with a lot of intensity and scored the goal," Xavi told Canal Plus.
"I think after that we stepped up to the challenge and deserved at least a draw.
"I think we played two good games and had our chances in both legs but it slipped through our fingers," he added ruefully.
"We can only congratulate all the Atletico fans."
Image: Thomas Mueller (R) celebrates after scoring a goal during their Champions League quarter-final second leg soccer match
Photograph: Michael Dalder/Reuters