The Cameroon international arrived during the summer transfer window as a replacement for long-serving goalkeeper David De Gea, having previously worked with manager Erik ten Hag at Ajax.
Praised for his ability to play from the back during his time at Inter Milan, Onana struggled during a difficult start to the season for United.
They are having their worst defensive run since December 1978 and are threatening to undo the work Ten Hag did to take the team through his first season in charge.
After a positive start for United at the Allianz Arena on Wednesday night, the 27-year-old allowed Leroy Sane’s shot underneath him to give Bayern the lead.
Serge Gnabry doubled their advantage less than five minutes later before Rasmus Hojlund gave the visitors hope.
Harry Kane would then add a penalty after Chrisitan Eriksen was adjudged to have handled inside the box before there was late drama.
Casemiro scored a second for United, only to see Mathys Tel fire home a fourth for the German champions, but the Brazilian then scored a third late on in the dying embers, although there was no time to come back.
“It’s tough,” Onana told TNT Sports.
“We started very well. After my mistake we lost control of the match. It’s a difficult situation for us, for me.
“The team was down because of that mistake. I have to learn from it and be strong. I’m happy with the team’s comeback.
“I have a lot to prove. My debut in Manchester wasn’t as good, not as I would like. It was one of my worst games. We have big ambitions.
“It’s a difficult time. We have to be together and work hard.”