Nico Williams, Yeray and Javier Martón struck for Athletic.
"We've done well in this league phase, our objective was to qualify," Valverde said after the match.
"To finish in the top eight was there and it means a lot to us. Getting through to the last 16 is a huge reward.
"After a long time without playing in Europe, with a lot of players who are new to these competitions, we've gone through and we're also still doing well in LaLiga."
Finishing in the top eight means The Zurigorri skip a play-off round and will face either AS Roma, Viktoria Plzen, Ferencvaros or FC Porto in the round of 16.
Valverde also said: "We don't care about the opponent. We're very excited to be back in Europe and we want to enjoy it. Having the second leg at home is important as any mistake can eliminate you.
"We're only going to have one match a week for a few weeks, which is good for training and resting. We've come through this well and let's see if we can catch our breaths a bit and be prepared for what's to come."