A 2-1 loss to Basel in Switzerland consigned the Red Devils to third place in Group C and a place in the Europa League. They have not failed to make the knockout stages of the competition since 2005.
"It's a big disappointment," Evra told reporters. "Since I played with Monaco, I've never been out in the first round.
"It's catastrophic and we feel very sad but we deserved to go out. It's not about tonight but throughout the competition - we threw away qualification.
"We created a lot of chances tonight and they get only two chances but they win - and deserved to win. From the beginning, we never played well in the Champions League."
Evra emphasised the importance of not letting the disappointment of their departure from the competition creep into their domestic form, however, as they look to keep up the pressure on Premier League leaders Manchester City, who they will also face in the FA Cup third round.
"If you play for Manchester United, you have to be strong," he said. "I think the gaffer knows the players will get back because they've got a strong mentality.
"For six years, as I said, I've never been out in the first round so it's a new experience for us. We have got to get back and get focused on the Premier League as that's the target - to win the Premier League.
"It will be more difficult for us [with matches in the Europa League on Thursdays] but we deserved it with the way we played. We deserve to play on Thursday and Sunday.
"But we are Manchester United. Today is a really bad day for everyone - the fans, staff and players. We must make sure we don't throw the season away.
"We still have things to win and must make sure we play with heart. The fans deserve something better than us throwing away the Champions League."
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