Arsene Wenger's surprise decision to start Yaya Sanogo ahead of Olivier Giroud appeared inspired as the pace of the French youngster and recalled Arsenal winger Alex Oxlade-Chamberlain unsettled Europe's tightest defence in the opening stages, only for Mesut Ozil to spurn a ninth-minute spot-kick.
Wojciech Szczesny then brought down Arjen Robben eight minutes before the break at the other end and was sent off, although David Alaba's resulting penalty struck the post for the second missed spot-kick of the game.
Following that, Munich laid siege to the Arsenal goal and they finally got their reward via a strike from the magnificent Toni Kroos, before substitute Thomas Muller put them firmly in control of the tie with a late second.
Szczesny will miss the return match in Munich and may also find himself in hot water with UEFA after appearing to make an offensive gesture when leaving the field after being shown his red card.
Bayern had stunned Manchester City with early goals in both their group-stage games against Arsenal’s Premier League counterparts and Kroos almost repeated the trick when he sent a 25-yard effort arrowing for the top corner in the second minute.
Szczesny, though, made a fine fingertip save and Arsenal went on to pin back their illustrious opponents during the opening exchanges.
Manuel Neuer stopped Sanogo from giving Arsenal reward for their early endeavour as he produced an excellent reaction stop following Jack Wilshere's prodded pass towards the striker.
Wilshere then released Ozil and Jerome Boateng - who endured a particularly torrid start - felled his Germany international team-mate in the area. But Ozil failed to capitalise from 12 yards as Neuer pushed away a tame effort.
That disappointment checked Arsenal's momentum, although Neuer had to rush out and deny Oxlade-Chamberlain in the 24th minute when the winger got the better of Alaba on the end of Sanogo's raking pass.
Arsenal left-back Kieran Gibbs departed with a hamstring problem after half an hour and replacement Nacho Monreal did not get up to speed quickly enough.
Robben raced past a static Monreal onto a delightful chipped pass from Kroos in the 37th minute and Szczesny charged out and brought the Dutchman down.
A lengthy delay followed and, although Alaba sent replacement Lukasz Fabianski the wrong way, his penalty rattled the base of the post.
Predictably, Bayern began the second half firmly on the front foot as Thiago Alcantara narrowly failed to connect with a cross from the left and Kroos drilled a low shot at substitute goalkeeper Fabianski.
Laurent Koscielny then stabbed goalwards from Ozil's 51st-minute free-kick but it was a moment of brief respite and the visitors led three minutes later, Kroos curling into the top corner from 20 yards.
Fabianski then saved twice from Robben and Muller almost found a decisive touch to cap the latest rampage from Rafinha in the 66th minute.
By now the attacks were relentless and when Muller stooped to head a deserved second with two minutes left it was no more than Bayern deserved.
They could even have added a third when Kroos rattled the post as Bayern looked to pile on the misery.
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