Leganes were beaten 3-2, with Gonzalo Garcia scoring an injury-time winner for Real Madrid.
Valverde said afterwards: “I'm pleased for the win and to progress in the cup. Until the second goal, we played well. We played the ball from one side to the other, moving quickly, looking for space and attacking. That's the attitude all players should have and we have to keep improving.
“When the team wins, the individual performance is secondary. I'm pleased with the victory and that I'm playing matches for this club.”
The win was Valverde's 200th triumph as a Real player, he added: “I'm pleased and proud to complete milestones at this club. This is part of my family, which helps me to be in better form every day. We all pull for the same side, we're all together and we're a team. I also want to thank the team because it's a family. They're there through thick and thin.”