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Valencia were denied a fourth consecutive win as Italian striker Simone Zaza had a goal ruled out for handball 17 minutes from time.
Sevilla's best effort saw Stevan Jovetic hit the post early on, but Jorge Sampaoli's men have now won just one of their last eight games in all competitions.
A point means Sevilla trail third-placed Atletico Madrid by three points in the battle for automatic Champions League qualification.
However, they enjoy a comfortable eight-point lead over fifth-placed Villarreal, who travel to Alaves on Monday.
Earlier, Leo Baptistao struck in stoppage time to hand Espanyol a 1-0 win at Leganes.