It was the hosts who threatened first at the Martinez Valero, but Enzo Roco couldn't find the target with his header early on.
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The opening goal came just before the hour mark when Barcelona loanee Gerard Deulofeu fired a low free kick into the area that fortunately ricocheted into the path of Bacca to bundle home from close range.
Sevilla put the game to bed 17 minutes from time when Denis Suarez raced away down the left and crossed for Gameiro to head home at the near post.
Elche, who are still without a league win at home this season, drop into the relegation zone and are just a point off the bottom.