United approached the home leg of their quarter-final tie against the holders as underdogs, but produced a determined showing to keep themselves in contention for a place in the last four.
Nemanja Vidic put David Moyes' side just before the hour mark, only for Bastian Schweinsteiger to level with a crucial away goal for Bayern, before being sent off for a second bookable offence in the 90th minute.
After the game, Bayern coach Pep Guardiola questioned United's defensive mindset, but goalkeeper De Gea believes the Premier League champions performed in the right manner.
"We played them in a proper way," the Spaniard said. "We didn't let them to play like they're used to and that's why they could be frustrated.
"I think it's important people don’t forget Manchester is a big club. Well, we are fighting against a team which I consider one of the best in the world, We had the chance to win, so we are satisfied."
And the Spaniard is hopeful that United can produce their best at the Allianz Arena and progress to the semi-finals.
"It looked like they would win easily but we gave our best," he added. "I think we made a good performance and we could even win the game. The tie is open and hopefully we will win there (Munich).
"This is football and our team has big hopes on going there and get the victory. Hopefully, we will be able to win."
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