Athletic won 3-0 on Saturday thanks to Oihan Sancet's hat-trick.
"We anticipated that Girona would try to escape our press," Valverde said after the match.
"It was difficult for us at the start, because it's not easy to play against them, but once we managed to steal possession in certain areas and get our midfielders on the ball, we started to create more danger.
"Our press was crucial to the win. They gave away the first goal, it hurt them and we stepped up a level to score two more goals. It was a good game."
On Sancet's performance, he continued: "When he's well, he's a decisive player for us. We need him to continue like this. After coming back from injury, he's been a very good addition to the team and we hope he continues to play at this level."
Maroan Sannadi and Endika Buján also featured for their Athletic Club debuts.
"Debuts are always exciting moments for players," Valverde added. "They have to settle properly to be able to play their game. Today was a good game to make a debut and they did well.
"They were brave and didn't go missing. But a debut is just one game. What this is really about is making sure there are many more matches after. That is what Athletic is. These players will get to grips with this level bit by bit.
"Athletic facilitates this, because our philosophy means that these players have that space. But then, they have to earn more time so they don't just become an anecdote."