The English side reached half time on level terms, but they were blown away during the second 45 minutes in Germany and now face the prospect of exiting the tournament at the second-round stage for the seventh successive year.
Arsenal went into the game with memories of a 5-1 thrashing in November 2015, and Arsene Wenger's side started as though they were already fearing a similar defeat on Wednesday evening.
Moments after Arturo Vidal had tested David Ospina from long range, Robben gave Bayern the lead in sensational style as he found the top corner of the net via the underside of the crossbar after being allowed to shoot with his favoured left foot from 25 yards out.
The home side did not have to get out of first gear during the opening 25 minutes but the calmness with which Bayern were playing changed after Manuel Neuer spilled a Mesut Ozil free kick and almost let Arsenal score an unlikely equaliser.
While the leveller did not come at that moment, it did three minutes later when Robert Lewandowski fouled Laurent Koscielny in the penalty area and although Sanchez saw his spot kick saved by Neuer, he kept his composure to fire a volley into the bottom corner.
The Premier League outfit rallied and five minutes before the break; Granit Xhaka volleyed straight at Neuer from 16 yards when he probably should have scored, and that led to Bayern regaining their rhythm just before half time.
Lewandowski and Mats Hummels both headed off target from no more than 10 yards out, but the visitors were lucky to escape when Mesut Ozil fired straight at Neuer from an acute angle after being played in behind the defence.
Soon after the restart, Arsenal were dealt a huge blow when Koscielny was forced off with an injury and it led to Bayern running riot in the final third.
Three minutes later, Lewandowski's clever flick released Thiago Alcantara behind the Arsenal defence and he produced what seemed a routine finish in drilling his shot into the bottom corner from 12 yards.
Lewandowski hit the crossbar, Robben saw a close-range effort cleared off the line by Kieran Gibbs's hand, and Ospina pulled off a wonder save to deny Javi Martinez as Bayern threatened to inflict a heavy defeat on their opponents.
They finally netted their third goal of a sublime 10-minute spell when Thiago's shot from the edge of the penalty area took a huge deflection to deceive Ospina and head into the opposite corner.
Bayern had further chances to extend their lead, with Vidal seeing another shot from distance saved by Ospina, before the Arsenal goalkeeper flung out a hand to stop Douglas Costa finding the far corner.
The fifth goal finally arrived with two minutes remaining as Muller found the bottom corner from 12 yards just seconds after being introduced as a late substitute.
Arsenal managed to reach full time without letting in a sixth, but they will take to the pitch at the Emirates Stadium next month effectively requiring a miracle to deny Bayern a place in the quarter-finals.