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United needed to overturn a 3-2 deficit from the first leg, but they never looked like troubling Marcelo Bielsa's slick side and went out after a one-sided 2-1 loss at San Mames saw them beaten 5-3 on aggregate.
It was the latest setback in a string of European disappointments for United, who meekly exited the Champions League at the group phase after losing 2-1 at Swiss champions FC Basle in December.
"It has been a bad year in Europe for Manchester United. We have had a difficult season," said Evra, who captained United in Bilbao.
"We have been more focused in the league, I don't know why. You have to be honest and say we don't deserve to be in the Champions League or the Europa League this season."
Beaten 3-1 by Barcelona in last season's Champions League final, United now find themselves without a major final to look forward to for the first time since the 2001-02 season.
Their only hope of silverware rests with the defence of their Premier League title.
Sir Alex Ferguson's side currently lead nearest rivals Manchester City by a point, ahead of their trip to relegation-threatened Wolverhampton Wanderers on Sunday.
