The French winger was moved into a more central role with striker Mario Mandzukic rested for Wednesday's Champions League quarter-final second leg against Real Madrid.
And Griezmann proved his versatility with just five minutes played, slamming an overhead kick past Deportivo goalkeeper Fabricio from 12 yards out.
That was never likely to be the end of the scoring for Griezmann, who tortured the Deportivo defence throughout with his outstanding pace, and so it proved.
Griezmann's second, a much simpler but equally effective finish on the half-volley, felt like a game killer even if there was still 70 minutes left to play.
From that point on, Atletico allowed Deportivo control of the ball but not of the game, as the visitors posed the only real threat with occasional breakaway attacks.
Oriol Riera did set up a grandstand finish with a powerful header at the near post in Deportivo's only really presentable opening, but they could not find the leveller they so craved.
Atletico pressed Deportivo back from the kick-off and were rewarded with just five minutes played when Griezmann sent a brilliant overhead kick into the roof of the net having been given the freedom of the penalty area.
Although the Frenchman appeared to be offside when the ball was flicked back into the area by Jose Gimenez, replays showed Riera's delayed run out of defence was playing him on.
Deportivo battled to get a foothold in the game, but after 21 minutes Griezmann doubled the visitors' advantage, driving a low half-volley back across goal after a throw-in from the left found its way to him at the far post.
Despite their dominance, Atletico showed a familiar devilment in their play, with Diego Godin booked for leading into a challenge with his elbow and Raul Garcia shown a yellow card soon after for dissent.
Deportivo dominated the ball for the remainder of the half, but could not force mid-week hero Jan Oblak into a single save of note as Atletico easily soaked up the pressure.
The theme continued in the second half, with Atletico surrendering possession without conceding chances, while Griezmann chased his hat-trick with a couple of threatening runs into the area.
With his thoughts firmly on the Champions League, Atletico coach Diego Simeone brought off Arda Turan and Griezmann for Raul Jimenez and Fernando Torres respectively.
The duo were involved in an extraordinary miss soon after coming on, although it would be unfair not to mention Deportivo goalkeeper Fabricio, who made a great triple save.
Saving Jiminez's initial shot when one-on-one, Fabricio then kept out efforts from Raul Garcia before Alberto Lopo cleared off the line from Torres.
And Atletico were given a nervous finale when Riera powered home a header at the near post after Sidnei's outstanding cross from the left with 12 minutes remaining.