The French forward, who has committed himself to La Real despite rumoured interest from Chelsea and Paris St Germain among others, netted twice in the opening 12 minutes.
Former Arsenal striker Carlos Vela got on the scoresheet shortly after half-time and Griezmann completed a sensational night with his third strike as Sociedad returned to winning ways in the Primera Division.
They had collected just one point from their last two games to slip away from the Champions League places, but an emphatic win puts them back in the picture.
Sociedad quickly settled into their stride and took the lead after four minutes when Vela played in Griezmann, who latched on to the long-range ball before excellently lifting over onrushing Antonio Tono.
The Elche goalkeeper was similarly powerless to prevent Griezmann giving the hosts a two-goal lead eight minutes later.
The 22-year-old Griezmann headed high into the top corner following a superb right-wing cross from Xabi Prieto.
To their credit, Elche refused to surrender and saw plenty of the ball in the first half but failed to create any meaningful chances.
That would come back to haunt them five minutes into the second half as Vela killed off any realistic hopes the visitors had of getting back into the match with a wonderful solo goal.
The Mexican wove his way to the edge of the area before unleashing a superb volley into the top corner.
Griezmann sealed his hat-trick in the 74th minute, netting from close range after Tono and failed to properly clear Imanol Agirretxe's cross as Sociedad moved within a point of fifth-placed Villarreal.