The Spanish side was unable to overcome the English team's strong squad depth, which was a key factor in the match.
Arsenal's substitutes Gabriel Martinelli and Leandro Trossard scored the goals that secured the win for the visitors.
Valverde acknowledged that the match was close, with both teams creating chances, but ultimately, Arsenal was more effective in their finishing.
He noted that Athletic Bilbao's high-pressing strategy was disrupted by the fatigue that set in during the game.
Valverde praised the team's overall performance, saying they were recognizable and competitive against a top-level opponent, but acknowledged that Arsenal's clinical finish was the difference-maker.
He also highlighted the impressive performance of midfielder Mikel Jauregizar, who has become a key player for the team in a short amount of time.
Valverde praised Jauregizar's ability to perform well in duels and his importance to the team's future.