The Gunners were 2-0 victors on the night thanks to an early Olivier Giroud strike and a late Laurent Koscielny goal, but after their 3-1 defeat in north London it was the German side who progressed on away goals.
Mario Mandzukic's late third goal was the difference between the two sides over the two legs and it is a goal which still irks Wenger.
"It's very difficult to go out of the Champions League for anybody, especially when you lose at home, you make it difficult for yourself," he told Sky Sports 2.
"We played against one of the best teams in Europe.
"The spirit and the performance was there we had a great go.
"My regret from first game is that we conceded a cheap goal in the last five minutes when we could lose 2-1 at home.
"You can see how important that goal was tonight [Wednesday]."
Despite his frustrations, a victory against a side 20 points clear in the Bundesliga did give some comfort to the Frenchman.
"Positives, of course, were the performance and the spirit. The clean sheet and overall defensively we were very good. We lost some balls off some situations which frustrated at times.
"I am confident we will give everything to come back next season. Our young players done very well lets finish well this season strongly."
The result brought back memories of last year's exit to AC Milan when they were 4-0 down in the first leg before winning the second 3-0.
"We have the same regrets as last year (against Milan)," Wenger said.
"The first game we have nerves but second game we show how good we are."
Arsenal often had their backs to the wall and had to rely on their back four and goalkeeper at times in what was an impressive defensive display.
Arsenal defender Carl Jenkinson spoke of his pride in his team-mates following their Champions League exit at the hands of Bayern Munich.
"We got that early goal we needed and we knew it would be important to keep it tight at the back," Jenkinson said.
"We did that, we got the second goal and I thought we could do it.
"I'm proud of the boys and proud of the effort we put in when everyone wrote us off."
Bayern winger Arjen Robben hopes the defeat will be a wake-up call for his side as they look to win the competition for the first time since 2001.
"We warned each other, we said 'don't let them come into the game, they have nothing to lose. One thing we don't want is them to score an early goal'," he said.
"In the first half we didn't play to our level, we weren't aggressive enough, we didn't win enough second balls and we didn't come into the game.
"All compliments to Arsenal, they did a great job, but we are happy that we are through."
"The second half we played better and we created chances bit more luck, sharpness we should score a goal.
"It wasn't our night in the end have to accept you are through and have to be happy but it is difficult.
"Lets hope its a wake up call for us, we have done amazing, we have played some terrific football .
"You always have to be focused on our own game, everybody was speaking about us saying it couldn't be any better but maybe (this result) could be good for the future."