Bayern made sure of their place in Europe's premier club competition with a 1-0 win in Zurich on Tuesday following their 2-0 first leg win at home last week.
"A huge weight has been lifted off our shoulders because this is where Bayern belongs," said Bayern sports director Christian Nerlinger.
"We have been feeling the pressure on the club and players in the past few weeks but we have now taken that crucial first step," Nerlinger told reporters.
Striker Mario Gomez's seventh minute strike quickly settled any nerves and put the game essentially out of the Swiss team's reach with the hosts needing four goals to advance.
"After our goal, we were not nervous any more because defensively we are a very good team and allow little to happen in the back," Heynckes said. "It may have looked easy but we played very good football."
Dutch coach Louis van Gaal was sacked late last season with the team's defensive operation looking more like Swiss cheese than traditional German steel.
"We wanted that early goal to settle our nerves and for the rest of the game we followed through exactly like we should do," said scorer Gomez.
The four-time European champions will now wait to find out their group opponents at the competition's draw on Thursday.
