Griezmann shrugged off the disappointment of finishing third behind Cristiano Ronaldo and Lionel Messi for world's best player at the inaugural Best FIFA Football Awards on Monday to force home the opener four minutes into the second half.
A fine individual effort from Marko Livaja gave Las Palmas a brief lifeline, but Angel Correa looked to have killed the tie off just after the hour mark.
Despite holding a commanding 2-0 first leg lead, Atletico boss Diego Simeone took no chances by naming a strong side.
Griezmann was included despite making the trip to Zurich to attend FIFA's gala in Zurich just 24 hours earlier and had the first big chance when he fired straight at Raul Lizoain on 23 minutes.
Las Palmas had chances to get back into the tie as Miguel Angel Moya produced a great save to deny Livaja's header at the near post before Tana blasted over when well-placed inside the area.
Livaja took advantage of some uncharacteristically lax defending from Diego Godin and Jose Maria Gimenez to drive through the heart of the Atletico defence and smash the ball into the far corner.
However, just four minutes later Correa controlled a long ball from Koke and shrugged off his marker before finishing from a narrow angle to put Atletico back in front.
And Garcia then volleyed home himself from Jonathan Viera's free-kick deep into stoppage time, but Atletico held out to progress.
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