Griezmann scored all four goals to reach triple figures for Atleti as a sixth straight league win increased the pressure on the Catalan giant ahead of Monday's top-of-the-table clash at Camp Nou.
With Barcelona due in action at Las Palmas on Friday (AEDT), Atletico needed a victory to cut the cushion enjoyed by Ernesto Valverde's side, and in-form Griezmann did the honours in supreme style.
The striker opened the scoring with a fine finish from Koke's superb throughball, having earlier been denied by the woodwork with a free-kick that curled a touch too much.
Griezmann scored his next set-piece though, blasting into the top-right corner to net his 101st Atletico goal and ensure a 0-0 draw at Butarque earlier in the season was forgotten.
The striker wrapped up his treble with a powerful header from a left-wing Filipe Luis cross and struck his fourth from a Diego Costa centre to give a rampant Atletico further hope ahead of its weekend visit to Barcelona.
A Costa snapshot required a save from Ivan Cuellar in the first minute and Saul Niguez thumped a drive just wide as Atletico hinted at the dominance to come by making a rapid start.
But Leganes should have taken the lead in the 11th minute, Javier Eraso missing from close range when he was found unmarked six yards out from a Gabriel Pires cross.
Atleti took heed of that warning and Griezmann was unlucky not to score with a cute free-kick that beat Cuellar but cracked the outside of the post.
Griezmann, having scored a hat-trick against Sevilla on, would not be denied when he ran through on goal after 25 minutes, though.
Koke's superb first-time pass bisected the Leganes defence, with Griezmann staying cool to hold off captain Martin Mantovani and finish past Cuellar with ease to bring up three figures for Atletico in all competitions.
Griezmann's earlier set-piece proved a sighter as he doubled the Atletico lead before the break.
From a similar position to the right of the D, Griezmann fired around the wall and into Cuellar's top-right corner, the goalkeeper unable to keep out the powerful strike.
Cuellar denied Griezmann his hat-trick before the half-time whistle, with Nordin Amrabat stinging the palms of Jan Oblak after the restart.
Griezmann was proving too hot to handle, however, with the striker completing his second treble in four days after 55 minutes.
The Frenchman dummied a pass from Costa to allow Filipe Luis to run clear down the left and deliver a measured cross for the former Real Sociedad star to thump home a bullet header.
Griezmann's 15th LaLiga goal of the season followed after 67 minutes, the striker beating Cuellar with a controlled volley from Costa's cross as Atleti ran riot.
Cuellar saved from Costa and Diego Godin's strike was correctly ruled out for offside, but Atletico's nine goals in its last two LaLiga matches is a sufficient sign the LaLiga title is not yet Barca's.