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 penalty when Gabi handled just inside the area.
The defeat ended Luis Enrique’s hopes of securing 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).
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.
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But Barca were a shadow of the team that conquered Europe so convincingly last season and have now lost three of the last four games in all competitions.
In addition, 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.
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On the rare occasions Suarez managed to work an opening he found Jan Oblak in fine form on a great night for Atletico.
Barca’s bad-tempered exit
The tie was littered with late challenges – there were eight bookings in total – while there were two flashpoints as Barca exited the competition on a bad-tempered note.
Suarez caught Uruguay team-mate Diego Godin with an elbow that left the Atletico defender with a swelling above his eye.
And another scuffle between the top two clubs in Spain broke out when Neymar appeared to lash out at Juanfran from behind.
Barca enjoyed 77% possession and had more shots than Atletico , yet could not disguise their frustration as they were undone by Griezmann’s double.
History repeats itself
This was the eighth meeting in 15 months between the sides, with Barcelona having won the previous seven.
Indeed, Atletico’s most recent win over their oppoents was at this stage of the Champions League two years ago.
Barcelona suffered a 2-1 aggregate defeat in 2014 with Diego Simeone’s side eventually reaching the final – only to lose 4-1 to Real Madrid after extra time.
Simeone will be confident after this fine performance that his side can go one better this time.
Messi remains on 499 career goals, just as he has done since 30 March when he scored for Argentina.
He has gone 452 minutes without scoring for Barcelona.
On Wednesday, Messi did not look like finding the net as he failed to register a shot on target in a Champions League game for the first time since 2014 – also against Atletico.
The 28-year-old cut a frustrated figure as Godin and Lucas Hernandez produced commanding performances at the heart of the home defence.
Messi’s last touch of the game was a free-kick on the edge of the area, when they should have had a penalty, which flew wide.