A 3-1 win on the night at the San Mames was enough to secure a 4-3 aggregate triumph for the La Liga club.
"When the game finished I was delighted for [the players] because they've put in an enormous effort and a massive commitment to deserve this victory," Bielsa told UEFA.com.
"I was also delighted for the fans because these games put an onus on the supporters. They want to be able to celebrate but there's also a fear of failure, yet they showed an appreciation and a gratitude for the players' effort."
Bilbao will take on fellow Spaniards Atletico Madrid in Bucharest on Wednesday, May 9.