The Gunners nearly overturned a 3-1 deficit at the Allianz Arena, but goals from Olivier Giroud and Laurent Koscielny could not stop the North London outfit exiting the competition on the away-goals rule.
Heynckes said that his team have learnt lessons from the match, insisting that they "can take nothing for granted".
"Overall, we'll be very critical when we analyse the match," the club's official website quotes Heynckes as saying.
"We hadn't planned on falling behind after three minutes. It played straight into our opponents' hands. We improved after half time and had better chances than Arsenal, but we didn't control the match and we weren't as composed as normal.
"We've had a scare but we got away with it. We've won nothing yet. As we've seen today, the Champions League is totally different. It's given all of us something to think about. You can take nothing for granted."
The draw for the quarter-finals will take place in Nyon, Switzerland tomorrow.