The result against the European champions ensures that David Moyes's team travel to Germany next Wednesday with a chance of reaching the last four, and De Gea has admitted that he was pleased to prove critics wrong for doubting United's capabilities.
The 23-year-old is quoted by UEFA.com as saying: "This is a big club and at difficult moments, you stand up to show people how big you are.
"I am pleased with that – people thought it was going to be easy for Bayern and we have produced a big performance. We are satisfied. We feel that we played really well. It was a big effort against a very strong opponent."
With the club currently 10 points adrift of the top four in the Premier League table, it is likely that United will have to win the Champions League in order to qualify for the competition next season.