Victory at the Ramon Sanchez-Pizjuan ensured the visitors are safe from relegation while helping Getafe, fourth, and Valencia, sixth, in their bid to finish in the top four ahead of fifth-place Sevilla.
Leganes broke the deadlock after eight minutes when Youssef En-Nesyri scooped an effort into the top corner from just inside the box.
Denmark international Martin Braithwaite doubled the visitors' lead after Sevilla failed to clear a long throw in the box.
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Leganes nearly scored again straight after halftime, with goalkeeper Tomas Vaclik making a good save to deny Juanfran.
With Sevilla struggling badly, Leganes wrapped up their impressive triumph when Braithwaite teed up Oscar Rodriguez for the third.
Getafe can move three points clear of Sevilla, who have 55 points from 36 games, if they beat Girona on Sunday, while Valencia can pull level with Joaquin Caparros's side with a win at Huesca.
"In the second half, without improving much, it is true that we tried to score but we were against the clock," said Caparros.
"They were comfortable, they do what they do very well and it's a pity because we lost a good opportunity.
"A defeat is never good but with how little there is to go, it's hard, (but) we'll fight for the six points and compete until the end."
(Reporting by Rik Sharma, editing by Pritha Sarkar)
