Trailing 5-1 from the first leg and striving to become the first team in Champions League history to overturn such a deficit, Arsenal led at halftime through Theo Walcott's goal.
But once Robert Lewandowski equalised from the penalty spot and Arsenal skipper Laurent Koscielny was sent off, the Bundesliga leaders ran amok against their dispirited hosts.
Robben capitalised on a defensive mistake in the 68th minute before substitute Douglas Costa curled in a third to send Arsenal's fans streaming towards the exits.
Bayern were not finished though and Arturo Vidal struck twice as Arsenal slumped to their worst home defeat since 1998 and a seventh successive elimination at the competition's last-16 stage in humiliating fashion.
Arsenal fans had mounted a small-scale demonstration calling for an end to manager Arsene Wenger's 21-year reign at the club before kickoff - yet there was plenty to admire from his team in the first half as they attempted mission impossible.
In his programme notes Wenger called for a salvaging of pride after Arsenal's second-half capitulation in Germany.
Alexis Sanchez was restored to the starting lineup after being dropped by Wenger against Liverpool and when Walcott smashed in a 20th-minute opener, it seemed Arsenal would at least exit with heads held high.
Olivier Giroud, pressed into action immediately before kickoff when Danny Welbeck injured himself in the warmup, wasted a great chance to make it 2-0 straight after the interval.
Then it all went horribly wrong for Arsenal and turned into a personal nightmare for Wenger whose position will again come under intense scrutiny once the dust has settled.
Bayern levelled on the night when Lewandowski was pushed over while running into the box by Koscielny in the 55th minute.
Referee Tasos Sidiropoulos reached for his red card, ending Koscielny's evening, and Arsenal swiftly collapsed, just as they did when the French defender went off injured in the first leg.
Lewandowski sent David Ospina the wrong way from the spot to knock the stuffing out of Arsenal. Robben then capitalised on some poor defending to put Bayern ahead before substitute Douglas Costa curled in a third in the 78th minute.
If it had been a boxing match the towel would have been thrown in but there was no hiding place for Arsenal and Vidal struck two late goals, both from close-range as the home defence went AWOL, to the obvious glee of Bayern's sizeable support.
Bayern fell just short of matching their 11-goal victory margin over two legs against Sporting Lisbon in 2009 when they won 12-1 on aggregate.
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Real Madrid's Sergio Ramos heads to score against Napoli in Naples REUTERS
Meanwhile, Real Madrid captain Sergio Ramos performed another rescue act to wreck Napoli's brave comeback attempt and take the Champions League title-holders into the quarter-finals on Tuesday after they had taken a first-half battering.
Napoli, trailing 3-1 from the first leg of the round of 16 tie, looked more than capable of overturning the deficit as they dominated the first half and took the lead through Dries Mertens.
But with Real wobbling, Ramos popped up to head an equaliser early in the second half and he produced another header six minutes later which deflected off Mertens and into the net.
Substitute Alvaro Morata added another in stoppage time against deflated opponents to give Real a 3-1 win that sent them into their seventh successive Champions League quarter-final by a deceptive 6-2 aggregate score.
Backed by a fervent crowd, Napoli made a fast and furious start and peppered the Real goal with shots in the opening 10 minutes.
They repeatedly carved open Real's defence with quick, clever angled passing and superb running off the ball.
One such move produced their opening goal when inspirational captain Marek Hamsik laid a perfectly-weighted first-time pass into the path of Mertens who lost his marker and placed his shot wide of Keylor Navas.
Real, fielding the same line-up that started last year's final which they won on penalties against Atletico Madrid, were being totally outplayed, although they nearly snatched an equaliser against the run of play.
Karim Benzema's pass sent Cristiano Ronaldo sprinting between the Napoli defenders and past Pepe Reina but the Portuguese forward fired his shot against the foot of the post.
Napoli also hit the woodwork with a shot on the turn by Mertens after Allan's pass was deflected into his path.
The hosts failed to begin the second half with the same intensity, however, and quickly paid the price after Hamsik uncharacteristically gave the ball away in midfield.
The move ended with Real winning a corner which Toni Kroos delivered perfectly for Ramos to rise above the Napoli defence and head the ball in to silence the crowd in the 51st minute.
Six minutes later, Ramos struck again from another corner, although this time the ball took a deflection of Mertens and Uefa awarded an own goal against the Belgian.
Faced with a hopeless task, Napoli completely lost their momentum and Real coasted through the rest of the game, Morata snapping up a rebound for the third goal.
(Mertens 24) Real Madrid 3
(Ramos 51, Mertens 57-og, Morata 90+1)
Real advance 6-2 on aggregate
(Walcott 20) Bayern 5
(Lewandowski 55-P, Robben 68, Costa 78, Vidal 80, 85)
Bayern advance 10-2 on aggregate
Barcelona v PSG
Dortmund v Benfica