The Spanish champions looked on course for back-to-back league defeats for the first time since April 2012 when Mikel Rico gave Athletic a deserved half-time lead yesterday.
But Griezmann levelled straight from the kick-off at the start of the second period and completed his hat-trick after Raul Garcia had put the visitors ahead from the penalty spot.
Victory moves Atletico back to within three points of Barcelona, who were 5-0 winners over Cordoba on Saturday, in second.
"This was my best match since I've been here. I need to keep on this path and keep working," Griezmann said.
At the new San Mames both sides were missing a number of key players with Aymeric Laporte and Ander Iturraspe banned for the hosts, while Atletico were without the suspended duo of Koke and Mario Mandzukic, as well as Miranda through injury.
Atletico failed to create anything of note in the opening 45 minutes, and could have been even further behind at the break had Aritz Aduriz decided to shoot when he was released clear on goal by Susaeta rather than trying to turn inside before being crowded out.
However, the champions emerged a different side after an inspired half-time pep talk from Diego Simeone and equalised straight from the kick-off as former Real Sociedad winger Griezmann came back to haunt his old Basque rivals once more with a precise header from Juanfran's cross.
