Barca led 2-1 from last week’s first leg at Camp Nou only for Atletico to level the aggregate score before half-time as unmarked France forward Griezmann headed home a Saul cross.
The 25-year-old then converted a penalty with only two minutes remaining after Andres Iniesta was penalised for handball.
Defeat ended Barca’s hopes of a La Liga, Champions League and Copa del Rey treble, while Ballon d’Or winner Lionel Messi has now failed to score in 452 minutes for the Catalans.
There were no such worries for Bayern Munich, meanwhile, who reached a fifth successive Champions League semi-final following a 2-2 draw against Benfica, which gave the German side a 3-2 aggregate win.
Mexican forward Raul Jimenez levelled the aggregate scores before strikes from Arturo Vidal and Thomas Muller gave Bayern breathing space. A Talisca free-kick drew Benfica level on the night.