Bilbao won last week's last-16, first leg 4-2 but were reeling at the San Mames when Xabi Etxeita put the ball into his own net soon after the break and Fabian Orellana converted a penalty to make it 2-0 and 4-4 on aggregate.
If Celta had managed another goal it would have been the first time in the 113-year history of Spain's domestic Cup that a team had overturned a 4-2 first-leg deficit.
However, Bilbao, whose 23 Cup triumphs are only bettered by Barcelona's 26, avoided conceding again to secure a last-eight clash against Malaga.
Coach Ernesto Valverde said the whistles were to be expected.
"We cannot be happy because we lost and qualification for the next round was in doubt," he added. "It was not a good match."
Five-times winners Sevilla romped into the quarter-finals when they thrashed Granada 4-0 at the Sanchez Pizjuan having beaten their Andalusian neighbours 2-1 in the first leg. Espanyol await in the last eight.
Villarreal also went through when they drew 2-2 at Real Sociedad for a 3-2 aggregate success. Sociedad coach David Moyes was sent off for the first time since taking charge of the club when he protested against a late offside decision.
Villarreal, who played the last 10 minutes with 10 men following the dismissal of defender Mateo Musacchio, will next play Getafe after the Madrid club beat Almeria 1-0 to advance 2-1 on aggregate.
"We are very pleased with the way the team is playing," Villarreal coach Marcelino said.
"We changed a lot of players compared with the league but the team has the same dynamic."
Atletico Madrid are on the verge of eliminating holders Real Madrid after winning the first leg at the Calderon 2-0.
The crosstown rivals meet at Real's Bernabeu stadium on Thursday and the winners of the tie will almost certainly meet Barcelona in the last eight as they take a 5-0 lead into their second leg at Elche.
Only top-flight sides reached the last 16 of the competition for the first time since 1969-70 when teams from lower divisions began competing.
(Reporting by Iain Rogers, editing by Ed Osmond)
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