Finely poised at 1-1 after the first leg in Naples, Bilbao were driven on by the fervent San Mames support and came back from a goal down early in the second half to secure a 3-1 victory on the night.
Napoli took the lead on the night two minutes into the second half through Marek Hamsik but Bilbao stunned the visitors with a ferocious spell of three goals in 14 minutes thanks to a brace from Aritz Aduriz and an Ibai Gomez strike.
The 4-2 aggregate win has provided the ticket for Bilbao's entry into the group stage of the Champions League for the first time in 16 years and Valverde hailed his team's ability not to panic, as well as the support from the Basque crowd.
He told Sport Mediaset: "The difference was made by the crowd at the San Mames, as they helped push the players on tonight and we kept faith in ourselves.
"Of course we kept calm, as we had prepared to face a difficult game against a great team like Napoli.
"This is an historic day for us, 16 years later we are going into the Champions League."
Two-goal hero Aduriz echoed his coach's comments about the importance of staying calm when behind, as Bilbao look forward to the draw for the group stage.
"I think the team worked very hard and we did not lose our heads after going a goal down at home," Aduriz told Sport Mediaset.
"It is one of our characteristics, that we never give up and never stop attacking.
"I think Napoli are a great team and they made life difficult for us, but we trusted in our own strengths and won out in the end."