Barca joined holders Sevilla in the last eight after the Andalusians won their second leg at Malaga 3-0 in the earlier kickoff to cruise through 8-3 on aggregate.
Deportivo La Coruna also advanced but came perilously close to elimination by second-division Cordoba before rallying to win 3-1 on the night and 4-2 over the two legs.
Bilbao held Barca to a 0-0 stalemate in last month's first leg at the Nou Camp and a hard-fought return match at the Basque club's San Mames stadium looked to be heading for extra time before the France international struck in the 75th minute.
Spain playmaker Xavi picked out Lionel Messi in the penalty area and Abidal streaked in from the left wing to sidefoot the Argentina forward's pass powerfully past Gorka Iraizoz, his first goal for the club since joining in 2007.
Abidal had started the game in central defence and did an impressive job of keeping Bilbao's burly Spain striker Fernando Llorente quiet before switching to his normal left back slot.
Freed from Abidal's attentions, Llorente latched on to Andoni Iraola's through ball and slotted past Jose Manuel Pinto in the 85th minute to give the home side hope but a spell of late pressure failed to yield a goal.
Barca coach Pep Guardiola said it had been an honour to play a match of such intensity at the San Mames, where Bilbao's passionate fans create a dauntingly intense atmosphere.
"The fans should be very proud of the Athletic players and they should be of the fans," he said.
"There is a great deal of merit in coming here and knocking them out on their turf in a tie that was extremely close and I am very happy with my players for managing it."
Madrid-based Getafe have the advantage for Thursday's return game after winning the first leg in Seville 2-1.
Sevilla, 2-0 winners against Atletico Madrid in last season's final, meet Villarreal or Valencia, who clash on Thursday after drawing 0-0 in Valencia.
Substitute Diego Perotti turned in a Jesus Navas centre in the 65th minute to effectively snuff out Malaga's hopes of overturning their 5-3 defeat in last month's first leg and Malaga defender Weligton was dismissed in the 72nd after picking up a second yellow card.
