Victor Fernandez's side went into Monday's fixture at Riazor having not scored a league goal since October.
But Luis Farina's 22nd-minute goal was enough for them to seal a crucial three points against bottom-placed Elche, with victory moving the hosts up to 16th in the table.
Farina curled a long-range strike into the top corner past Przemyslaw Tyton to earn the win.
Deportivo could have doubled their advantage early in the second half save for some rain-affected conditions, as Ivan Neves' ball through to Jose Verdu stopped dead in a puddle.
Elche could have denied their relegation rivals late on, but Faycal Fajr's on-target shot was stopped by the face of Deportivo keeper Fabricio.
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