There was a blistering start to the European tie - a year on from their last encounter in north London - and the holders might have snatched an early lead when Toni Kroos fired in a 25-yard effort towards the top corner, which Szczesny tipped over.
Yaya Sanogo had been a controversial selection in attack ahead of France striker Olivier Giroud, whose off-field actions have made headlines in recent weeks. However, the summer free transfer from Auxerre - who has been out for several months with a back problem - had a close-range shot palmed away by Manuel Neuer, before the German keeper saved a tame penalty from Oezil on eight minutes.
There was more drama just before the break as Arsenal goalkeeper Szczesny was shown a red card for bringing down Arjen Robben in the penalty area, only for full-back David Alaba to side-foot his spot-kick against the base of the post.
Bayern, though, went ahead on 54 minutes thanks to a superb 20-yard strike from Kroos, which whipped into the top corner past substitute keeper Lukasz Fabianski. The hosts tired heading into the final 20 minutes as Robben exploited the space in front of a jaded backline. Eventually, the pressure told, as substitute Thomas Mueller ghosted into space to score a late second goal to all but kill off the tie ahead of the return leg at the Allianz Arena next month.
"Arsenal were much, much better than us in the first 10-15 minutes, after the mistake from Oezil the game was equal," said Pep Guardiola, the Bayern manager. "From that moment we played better and after the red card for the goalkeeper the game was for us."
It had all started so promisingly. After that superb save from Szczesny to turn over that blistering strike from Kroos - a reported summer target for Manchester United, though why his club would ever let him go is unclear - which was heading into the top-right corner, Arsenal went on the offensive.
Sanogo's close-range shot was palmed away by Neuer and the goalkeeper also stopped an angled drive from Santi Cazorla before the home side were awarded a penalty in the eighth minute. Oezil's brilliant reverse stepover bamboozled Jerome Boateng, who upended the Arsenal playmaker and was booked. However, the penalty from the German was tame, nothing more dangerous than a light chip down the middle following a two-step run-up, and Neuer - a childhood friend of Oezil - saved.
"I prefer people to run properly at the ball," admitted Wenger. "Everyone has his own style and you have to respect it. There's not one way of penalty taking, if he's comfortable taking it like that . . . it's his style. It was a very difficult night but as well a night the team has shown some quality and some class."
The England winger Alex Oxlade-Chamberlain was then inches away from connecting with a flick-on from Sanogo, which had Neuer racing out to the edge of the Bayern area.
Arsenal were forced into a change on 31 minutes when left-back Kieran Gibbs hobbled off with what looked like a hamstring injury and was replaced by Nacho Monreal.
Bayern continued to probe and Robben's low shot was blocked by Per Mertesacker following another patient build-up down the left flank.
Then, in the 37th minute, the Dutch forward latched on to a chip into the Arsenal box, which he touched past Szczesny and was brought down. Italian referee Nicola Rizzoli had little choice but to award a penalty and show the keeper a red card. It was the 100th accrued by Arsenal under Wenger's managerial reign.
Szczesny made an offensive hand gesture as he trudged off down the tunnel and, after a considerable delay as the unfortunate Cazorla was replaced by Fabianski, Alaba struck the base of the left-hand post.
Bayern made the breakthrough in the 54th minute when captain Philipp Lahm fed Kroos on the edge of the penalty area, and as the Arsenal defence backed off, he cracked a fine curling shot into the top-right corner.
Arsenal tired heading into the final 20 minutes, and Robben exploited the space to curl a 20-yard effort just wide before Mario Goetze's powerful header was deflected behind.
Wenger made a change for the final quarter-of-an-hour as Tomas Rosicky replaced Oxlade-Chamberlain, but it was another substitute, on the opposing side, who was to have the final say.
Mueller first looked to have been tripped by Laurent Koscielny in the right side of the penalty box, but the officials waved play on.
Then, the killer blow. Lahm's cross from the right was headed in by Mueller in the 88th minute. And it could have been worse, but Kroos' low shot struck the base of the post in stoppage time, keeping the score at 2-0.