Despite Bayern going down 3-2 on the night, it was Robben's effort that ultimately proved United's downfall and it was a strike worthy of winning any clash as they tied up 4-4 on aggregate.
Franck Ribery's corner picked out the Dutchman who let fly with a left-footed volley to leave Edwin van der Sar with no chance in the United goal and bringing up Bayern's priceless second away goal of the contest.
The Red Devils looked set to storm through to the last-four of the competition after a blistering start in Manchester.
With Wayne Rooney making an unexpected return to the starting line-up following an ankle injury picked up in the 2-1 first leg defeat in Germany last week, it lifted the United faithful as they made an unbelievable start.
Darron Gibson opened the scoring inside three minutes, crashing home from distance as the English champions stamped their early authority on proceedings.
Rooney's lay-off was gathered by Gibson and he fired low past an unsighted Hans-Jorg Butt in the Bayern goal.
Just four minutes later, Nani's nonchalant flick doubled their advantage to send the crowd delirious once again.
Valencia's cross from the right was perfectly weighted for the winger to delicately flick across Butt and into the far corner, putting United firmly in the driving seat.
Rooney looked to be feeling his problematic right ankle soon after but Nani superbly swept home inside the area in the 41st minute as the Germans continued to struggle with United's free-flowing attacks..
It was another Valencia centre from the right that found the Portugal man and he kept his cool to slot high past Butt.
However, just two minutes later, Ivica Olic created a chance out of nothing as he squirmed through Michael Carrick's weak challenge before squeezing the ball home from an acute angle.
United survived another couple of scares as referee Nicola Rizzoli brought a frenetic opening 45 minutes to a close.
Just five minutes into the second-half, young United full-back was shown his second yellow card for a tug on Ribery's shirt and, just five minutes later, a struggling Rooney was finally replaced by John O'Shea.
His dismissal proved telling as the Germans upped the ante and United rarely threatened with ten-men as Bayern pressed.
United stood firm in the face of adversity, keeping Louis Van Gaal's men to striking from distance, as they threw bodies on the line to defend the narrowest of leads.
But it was a moment of magic that won the contest as Robben's fierce effort with 15 minutes remaining rocketed in, leaving his compatriot in United's goal helpless.
Dimitar Berbatov and Ryan Giggs were brought on but it was too little too late as the classy Germans made their numerical advantage count to set up a semi-final date with French side Lyon, who lost 1-0 at Bordeaux, but secured a 3-2 aggregate success following their 3-1 victory last week.