Atletico, the 2014 runners-up, needed only a draw to progress and their comfortable victory at the Calderon stadium put them level on 10 points at the top of the group with Benfica and ahead of the Portuguese club on goal difference.
Benfica, who host Atletico in the sixth and final round of games next month when top spot will be up for grabs, fought back from two goals down to draw 2-2 at Kazakh side Astana earlier on Wednesday.
Galatasaray have four points in third place, with Astana on three in fourth.
Atletico are appearing in Europe's elite club competition for the third season in a row for the first time and turned in another solid display in the Spanish capital.
Griezmann, who got both goals in September's 2-0 win at Galatasaray on matchday one, put Diego Simeone's side ahead in the 13th minute when he was left unmarked and headed a Gabi cross firmly past goalkeeper Fernando Muslera.
Gabi was again the provider for the France forward's second in the 65th minute, a tap in from close range that snuffed out any hopes of a Galatasaray comeback.
(Writing by Iain Rogers, editing by Ian Chadband)
