Mikel Rico gave the home side the lead after 17 minutes but Diego Simeone's men responded in ruthless fashion to keep their title challenge firmly alive
The former Real Sociedad man scored three goals in four La Liga games against rivals Bilbao prior to joining the Spanish champions and continued that streak as Atletico came from a goal down to end 2014 on a high.
The Champions League runners-up looked set to end a famous year with disappointment as Mikel Rico's first goal of the season handed the hosts a deserved early lead.
However, Ernesto Valverde's men failed to build on that in the remainder of a first half they largely dominated and were made to pay when Griezmann equalised less than a minute after the interval.
A slick Atleti move ended with Juanfran's delivery, allowing the France international to head the leveller, with Raul Garcia tucking home a penalty soon after.
Diego Simeone was without suspended pair Mario Mandzukic and Koke, but the lively Griezmann offered two further reminders of his quality to ensure Atleti's first win in four matches and maintain pressure on Barcelona and Real Madrid above them.
Both sides were guilty of giving away possession in a fast-paced start, with a spat between Iker Muniain and Jose Gimenez proving the main talking point of the first 10 minutes.
Gimenez - named ahead of Miranda in the centre of defence - and Muniain were involved in a number of off-the-ball incidents early on with goalscoring opportunities at a premium.
Instead, fouls and stoppages disrupted play until Mikel Rico marked his 50th La Liga appearance for Bilbao with a well-taken header.
The midfielder glanced home Markel Susaeta's free kick to open the scoring after 17 minutes, leaving Simeone visibly frustrated on the touchline.
Shorn of three key players, the champions looked out of sorts going forward as Bilbao's defensive pairing of the fit-again Xabier Etxeita and Carlos Gurpegui coped well with the threat of Griezmann and Raul Garcia.
Athletic - missing Aymeric Laporte and Ander Iturraspe due to suspension - could have made it 2-0 seven minutes before the break when Susaeta played Aritz Aduriz in on goal, but the striker was unable to produce a finish.
With the words of Simeone ringing in their ears, Atletico began the second half with added aggression and had their reward when Griezmann finished a swift break with a smart header.
They built on that momentum when a rather soft penalty given against San Jose allowed Raul Garcia put Atleti in front after 53 minutes.
Griezmann continued to prove a danger, adding his side's third from Gabi's long ball after a costly Gurpegui slip, before completing his first Atleti hat-trick with a close-range finish nine minutes from time.
It leaves Bilbao with work to do to regain a spot among their fellow European hopefuls in 2015, while Atleti stay in the hunt to defend their title going into the mid-season break.