The 3-1 triumph at San Mames saw Athletic reach the quarterfinals 4-3 on aggregate. Nico Williams struck twice, with wing-back Yuri Berchiche also scoring.
Valverde said afterwards: "There are games where your leaders have to stand up. We have those players, and it’s crucial. Nico feels a responsibility, and he tried from the first minute. He scored two goals – one of them crucial just before half-time – and hit a post too."
Athletic were handed an early advantage after Mats Hummels' 11th minute dismissal.
"When you have an extra player, you have to know how to play with it," continued Valverde. "We had to produce more crosses to find gaps with our passes. We pushed on, and we had chances – we had Nico’s against the post, Maroan's chances, and then everything went well."
Berchiche also said: "We had to be patient, to play the ball from one side to another, to tire Roma, and then try to get a goal through crosses and through balls. The goal just before half-time hurt them a lot.
"It’s my first goal of the season. This year I’ve taken my time to get one, and I picked the best moment.
"I want to thank them because they make our job easier. We’ve come from a difficult moment, but today they set that aside and carried us on their shoulders. We thank them from our hearts. The atmosphere was spectacular. I’ve lived some marvellous moments here, but today was something special."