Arsenal won 2-0 thanks to goals from substitutes Gabriel Martinelli and Leandro Trossard.
Valverde said afterwards: "It was a fairly even match. We knew they are very strong in individual duels, which is something we’ve worked on. We tried to press them high.
"It was a tense game with some disjointed play and we focused on making the most of the chances we had. They caught us on Martinelli’s move and the goal hurt us. They have strength and we felt the fatigue.
"We were recognisable on the pitch, showing what we are. It’s a shame we couldn’t give the fans a victory with the atmosphere, which was incredible. You always want the result to go your way and we tried, but this is the Champions League.
"We are in top-level competition and you have to accept things as they are. What matters is standing up to these kinds of teams and playing on equal terms against such opponents. Today they were more clinical, but I believe we performed at a very high level."
Valverde also highlighted the performance of midfielder Mikel Jauregizar, adding: "He has become a key player in just over a year. You could see his potential and he is surpassing that initial impression we had. He performs well in duels and is hard to get past. He is essential for Athletic and for the club’s future."