Goals from Mikel San Jose, Oscar de Marcos and Ander Herrera sealed the Basques' place at Europe's top table for the first time since 1997.
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 after their 3-1 win over Sevilla last week.
After kickoff was delayed by almost 15 minutes when home fans littered one of the goalmouths with toilet paper and ticker tape, 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 halftime.
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, which has three matches remaining but can no longer overhaul the Basque outfit due to its inferior head-to-head record.