The Premier League side, trophyless since their 2005 FA Cup triumph, were written off after a 3-1 defeat in London by the slick Bavarians.
But rather than roll over in Germany, Arsenal stunned nervy Bayern with Olivier Giroud's early goal at their Allianz Arena fortress and after Laurent Koscielny headed a second with four minutes left, Bayern were forced to cling on to reach the last eight.
"I believe we were very, very close," Wenger said.
"The spirit and the performance was there and we had a great go. The overall defensive performance was very good. They defended very very well and we lost some balls offensively in situations.
"It was frustrating sometimes but I put that down to the fact that they defended well and when we got the second goal, you could see that if we had put some more balls in the box we would have score another goal."
No team in the Champions League has recovered to qualify after losing the first leg at home by a two-goal margin and Arsenal were knocked out in the last 16 for the third season in a row.
Wenger pointed to Bayern's late third goal at The Emirates last month, scored in the 77th minute by Mario Mandzukic, as the hammer blow.
"The regrets we have tonight of course is from the first leg when you think that we conceded a cheap goal. Had we lost 2-1 at home you see how important that goal was tonight," he said.
Arsenal were without influential Jack Wilshere, Lukas Podolski and Bacary Sagna and keeper Wojciech Szczesny was rested on Wednesday but Wenger took pride from his side's display in defeating last season's beaten finalists.
"I know these players. I have said many times they have a fantastic attitude and spirit and we have the quality. But unfortunately we have to go out of the Champions League and it hurts very much."
Arsenal almost pulled off a huge comeback in the Champions League round of 16 last season, losing to AC Milan 4-0 in the first leg and narrowly going out after winning the return 3-0.
"We have a bit of the same regrets we had last year in the last 16 - first game nerves and afterwards, we have no choice and show how good we are," added Wenger. "This is a 180 minute game."
Arsenal now face a battle to secure Champions League football next season by finishing in the Premier League top four. With 10 games left, they sit in fifth place, five points behind Chelsea in fourth.
"I'm confident that we will give absolutely everything to be back next season and keep the basis of this team," Wenger said.
"You could see that the young players that played tonight are doing very well, let's finish well this season."
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