The only goal of the game arrived just before the half-hour mark, as Kevin Gameiro turned the ball home from close range after a corner was not dealt with.
Victory for Atletico extends their unbeaten streak in away matches to 17 and lifts them up to third place in the league standings, two points in front of Real Madrid who host Eibar later today.
Jan Oblak was forced into making an early save to keep Atletico level in a match that had been threatened with postponement due to wildfires in the area.
Atletico, without a win in their last two prior to today, got the breakthrough they craved when Gameiro fired home from close range for his first goal of the season.
Celta headed into the match without a loss in their last four matches and they very nearly found an equaliser in the second half, only for Oblak to produce another big save 52 minutes in.
Griezmann squandered a further chance later in the match to add to his tally for the season, but in the end it did not matter as Los Rojiblancos held out to get back to winning ways.