Renan Lodi scored the only goal at Old Trafford as Spanish champions Atletico advanced to the quarter-finals 2-1 on aggregate.
"We are really disappointed with the result. It's difficult to put into words how we feel now at the moment. We did not do enough in both games to win it. It's a very disappointing day for us," De Gea said after the match.
"They are a team with a lot of experience. They know how to play these Champions League games.
"They scored a goal, then they are a tough team so it was hard for us to create chances. We are out of one of the most important competitions so we are very sad."
United have reached the last eight of Europe's top club competition just twice since Alex Ferguson retired in 2013. The club's last trophy came when they won the Europa League five years ago.
"Of course it's not good enough. It's hard for the club, for us and the fans. It's really hard," said De Gea.
"This is where we are at the moment. It's a difficult situation. We have to keep fighting. I feel very sad. But we were not good enough."