Heynckes believes his team fell well short of their own expectations and this must be a lesson if they are to challenge for the Champions League, reports Goal.com.
"I must say my players are leaving here with a black eye," said the 67-year-old.
"I think I know European football, having had 150 games as coach in European Cups and I had warned my players."
Bayern went into the game having won 3-1 at the Emirates in the first leg last month, but goals in the third and 86th minute almost saw one of the most in-form teams in Europe exit at the last-16 stage.
"Arsenal have quality, they came to us wounded from the first leg. They were very motivated, driven and they got that early goal,"said Heynckes.
Olivier Giroud's third-minute tap-in was the perfect start for Arsenal as they set out to score three goals, a daunting challenge in Munich. Heynckes had urged his players to evade an early strike.
"That was exactly what we needed to avoid. To go into the lead in the third minute makes it so much easier for one team and more difficult for the other," Heynckes said.
Laurent Koscielny's 86th strike was not enough to send Arsenal through, but restored some pride for the Gunners after their defeat in the first leg.
Bayern are 20 points clear at the top of the Bundesliga and are favourites to lift the German Cup after reaching the last four, but need to perform better than they did last night to avoid disappointment in their remaining fixtures.