Ernesto Valverde's men knew a win against their mid-table hosts would seal a place in the competition for the first time since finishing second in the 1997-98 campaign.
Any nerves were settled within half an hour, as goals from Mikel San Jose and Oscar de Marcos put the visitors in control heading into half-time.
Both efforts came from left-wing corners as Vallecano failed to clear their lines, with Bilbao fans particularly exuberant following De Marcos' strike, pouring forward in the stands to greet the players, causing a protective barrier to collapse.
It appeared that nobody was seriously injured, and the visiting supporters' mood was improved in the second half when Ander Herrera squeezed a free-kick between goalkeeper Ruben's left-hand post to add gloss to the scoreline.
The result leaves Bilbao nine points clear of fifth-placed Sevilla, who have three matches remaining but can no longer overhaul the Basque outfit due to their inferior head-to-head record.
After kick-off was delayed by almost 15 minutes when home fans littered one of the goalmouths with toilet roll and ticker tape, San Jose gave Bilbao the lead in the 20th minute.
As Markel Susaeta curled in a corner from the left, Ruben came to collect but got nowhere near the cross, allowing San Jose to flick the ball home into an empty net with his heel.
The visitors doubled their lead 10 minutes later, De Marcos lashing the ball under Ruben after the hosts had failed to deal with another corner.
Herrera provided the icing on the cake 16 minutes from time, bending home a set-piece from 25 yards to spark scenes of jubilant celebration among the players at the final whistle.
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