The hosts made a perfect start to the match when they broke the deadlock after just three minutes, Griezmann lifting a shot over Tono having been found by a fine Vela pass.
Griezmann doubled his personal tally 10 minutes later, this time showcasing his heading ability by planting the ball in the top corner following a cross from Xabi Prieto.
The visitors had a decent chance to get themselves back into the contest five minutes after the break when Aaron Niguez forced a corner, but Botia could only steer his close-range effort wide from the resulting delivery.
The hosts took full advantage of that miss to score a third moments later as Vela joined Griezmann on the scoresheet with a stunning strike into the top corner from the edge of the box.
Griezmann completed his hat-trick with 15 minutes to play when Tono failed to deal with a ball into the box, allowing the French striker to convert from close range.
They came even closer in the 89th minute when Damian Suarez rattled the top of the crossbar with a free kick, but they were unable to get on the scoresheet as Sociedad cruised to a comfortable victory.