The German side struck twice in four minutes in the first half at the Allianz Arena to set them on their way to a 4-3 victory in their Group A opener.
A mistake by Cameroon international Onana saw the Bundesliga side take the lead when he let winger Leroy Sane's tame shot slip past him and into the net.
"It's difficult to lose this way because I feel in the beginning, we started very good. After my mistake, we lost the control of the game," Onana told TNT Sports.
"It's a difficult situation and for me especially because I'm the one who let the team down. But the team was very good; I think it's because of me that they didn't win this game.
"We just have to move on. This is the life of the goalkeeper and, if we didn't win today, it is because of me."
United manager Erik ten Hag said Onana, who moved to Old Trafford from Inter Milan in July, had the mindset to bounce back.
"(In) football on the whole, (there) will always be mistakes, and this is one mistake," the Dutchman told reporters.
"So, we also don't make it bigger than it is. He will bounce back, and he is a very important player."
