United, who this time last year were well on their way to the Champions League final, were outplayed in both legs by the Spanish club, losing 2-1 in Spain and 5-3 overall.
United forward Wayne Rooney scored a superb 80th-minute consolation goal but the damage was already done as the enthusiasm and skill of La Liga's seventh-placed side proved too much for the English champions.
"I don't think we can complain," United manager Alex Ferguson told Channel Five television. "There are always lessons in every football match whether you win or lose.
"It's the disappointment more than anything that we haven't progressed in the Champions League and we're out of the Europa League now."
He later added: "Trying to analyse this season against others is a bit difficult. There is a route in terms of some of the goals we have lost in these tournaments. They have been pretty poor. It has been a disappointing year.
"In the second half in particular, Bilbao were the better team.
"Their first goal was a bad one for us to lose. You can't believe an English team would lose a goal like that to a long ball. Ours was too late to do anything about it.
"Athletic could do very well. What we have seen, tonight and last Thursday, is a team whose workrate is higher than anyone I have seen in Europe.
"That takes you a long way but it is not just about their workrate. They have some fantastic qualities also.
"I would like to wish them well. I hope they do win it because it is a testament to the hard work they all put in, from their coach right through.
"They have created a cause. Such energy and determination to win football matches is an incredible thing to see. They deserve to win it."
Athletic coach Marcelo Bielsa said: "Victory against a team like Manchester United cannot be ignored.
"But I don't have any reason to think it has happened because English football is too insular."