Bastian Schweinsteiger was on target for Bayern 10 minutes after the break as he collected Franck Ribery's cross to fire home a close-range effort for his first goal since October.
Lukas Podolski hit back for the visitors two minutes later, taking on the Bayern defence and beating the outstretching Manuel Neuer as he drilled his shot home from a narrow angle.
Laurent Koscielny brought down Arjen Robben in the box after a tussle as the game approached full-time, but Thomas Muller's penalty was held out by the feet of Lukasz Fabianski before the ball spun just in front of the line, with the Arsenal goalkeeper making it safe seconds later.
By that stage, though, Arsenal's hopes of reaching the quarter-finals were bleak.
Oxlade-Chamberlain told ITV: "In the first game, losing 2-0 at home (meant it) was never going to be an easy task coming here. We went down to 10 men at the Emirates (after goalkeeper Wojciech Szczesny was sent off) and up until that point we were doing really well and it was an uphill climb.
"On the day - 2-0 - it wasn't the best result for us and we knew we had the chance to come here tonight and give it our all and try to come away with a result."
Bayern looked the most likely to break the deadlock in the first half, but after the Gunners struck back, Oxlade-Chamberlain believed the visitors were still in with a chance.
He added: "Bayern are always going to have weaknesses, just like any other team, and when you score a goal it puts anyone under pressure and (makes) anyone a little bit nervous.
"So we knew we needed something like that to make them uneasy and obviously when we did that we thought we had a chance and it wasn't to be in the end."