Nico Williams and Oihan Sancet struck as Athletic won on the night 2-0 and reached the semifinals by the same scoreline on aggregate.
Valverde said, "What I liked most about my team is the determination with which they came out from the first minute.
"This team has the possibility of reaching a final. Today was a very special day for us. We knew our fans would be fully behind us. By connecting with our people, we are a powerful team.
"Our idea was to make the most of playing at San Mamés. We needed to set a high tempo to the game, they were defending well. We had chances to score, but it was in the last play that we took the lead. We managed to maintain that advantage."
On facing Manchester United
Athletic will now meet Manchester United in the semifinals, with Valverde admitting playing the second-leg at Old Trafford is a disadvantage.
He said, "It’s going to be a tough match, with the second leg away.
"They have the Europa League as a route to the Champions League. It changes our perspective a bit from what we've experienced in these two rounds, but we’re going to play a strong match."