Bilbao travelled to Ukraine with just one point in their previous four UEFA Champions League matches.
But Valverde's men climbed off the bottom of Group H with a professional road win - highlighted by Mikel San Jose's smart finish in the 68th minute.
The defender's goal - and BATE Borisov's 3-0 loss to Porto in Belarus - means Bilbao sit third in the pool ahead of matchday six, a position that would see them drop into the UEFA Europa League.
And with qualification for the Champions League's last 16 beyond them, Valverde was pleased that his side's continental run may continue.
"I don't know if it's our best game of the year," he is quoted as telling AS.
"We've been brighter in others, but we had to work hard because we had to force it if we were to have options going forward.
"No matter what happened in Belarus, our only goal was to win.
"When a team is in last place, which we were before play, you have to win.
"We have done that and we're happy."
Bilbao host BATE in their final Group H clash in December, knowing if they avoid defeat they would secure theira spot in Europe's second-tier competition.
Get the best features, fun and footballing frolics straight to your inbox every week.
Thank you for signing up to Four Four Two. You will receive a verification email shortly.
There was a problem. Please refresh the page and try again.