The result left Sevilla fourth on 63 points from 35 games, six more than fifth-placed Villarreal with three matches left. Bilbao stayed ninth on 48 points after a blow to their hopes of clinching a Europa League berth.
Midfielder Ever Banega inspired Sevilla's fightback after they fell behind to an Ander Capa goal in the first half, having equalised in the 69th minute before he set up Munir El Haddadi's winner four minutes later.
Right back Capa fired Bilbao ahead against the run of play in the 29th minute when he side-footed home a first-time shot after Mikel Vesga's wayward effort was deflected into his path.
Bilbao keeper Unai Simon denied Suso and Lucas Ocampos with two superb saves but he was powerless when Banega curled a delightful free kick into the top corner from 20 metres.
Substitute El Haddadi turned the match on its head shortly after coming on, heading home from close range after Banega whipped in a fine inswinging cross from the right.
Sevilla keeper Yassine Bounou kept the visitors ahead when he clawed a stinging Iker Muniain shot out of the bottom left corner five minutes from time.
Sevilla host relegation-threatened Mallorca while Bilbao visit 12th-placed Levante on Sunday.
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