Barca arrived in Spain's capital with a 2-1 advantage from the first leg but poor defending allowed the unmarked Griezmann to power home Saul's cross.
Saul headed against the bar as Barca toiled before Griezmann scored a late penalty after Andres Iniesta handled.
Barca were denied a late penalty when Gabi handled just inside the area.
The defeat ended Barca's hopes of a La Liga, Champions League and Copa del Rey treble, while Atletico join Bayern Munich, Manchester City and Real Madrid in Friday's semi-final draw (11:30 BST).
Relive the action from Wednesday's Champions League quarter-finals
Barcelona's players surrounded the referee after Gabi clearly handled inside the area though he was standing outside it, the referee awarding only a free-kick.
But Barca were a shadow of the team that conquered Europe so convincingly last season as Lionel Messi failed to score for the fifth game in a row for his club, and Neymar and Luis Suarez were frustrated by excellent home defending.
On the rare occasions Suarez managed to work an opening he found Jan Oblak in fine form on a great night for Atletico.
MATCH ANALYSIS
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