The La Liga outfit beat United 5-3 on aggregate - 3-2 at Old Trafford, 2-1 at San Mames - to progress to the quarter-finals and knock the Reds out of European competition for a second time this season.
"There are always lessons to learn from every football match, whether you win or lose," Sir Alex told Channel 5. "It's disappointing more than anything. We haven't progressed in the Champions League and now we're out of the Europa League. I think the best team went through. I don't think we can complain."
The Reds travelled to Bilbao targetting a 2-0 victory but those plans were shattered when Fernando Llorente volleyed home on 23 minutes. The goal, albeit brilliantly executed, came via a long ball from the Athletic left back. In many ways it was typical English route one football, and it left Sir Alex bitterly disappointed.
"They got a really soft goal to put them ahead and that's the last thing we wanted. I can't believe we lost a goal like that. It was a really poor one.
"Otherwise, I thought it was quite even in the first half. I thought we did quite well. But at the start of the second half Bilbao did very well. They could have scored two or three goals.
"Towards the end we had a bit of impetus - we scored a goal but we didn't get anywhere near to winning it."
United, out of Europe and the FA Cup, now have only the Barclays Premier League trophy to play for. And although some fans will claim the Europa League exit is a blessing in disguise on that front,