Griezmann put the hosts ahead on away goals with a fine first-half header before converting from the penalty spot two minutes from time after Andres Iniesta had handled inside the area last night.
Barca, meanwhile, were left to rue referee Nicola Rizzoli's decision to give a free-kick on the edge of the box in injury time instead of a penalty when Gabi appeared to be inside the area as he was penalised for handball.
"This group displays values that are less and less common in society. We can win or lose, but we believe in these values that are important in life and once again we have won a tremendously difficult game."
Defeat is Barca's third in four games after a 39-match unbeaten run stretching back six months and sees their quest to become the first side to retain the trophy in the Champions League era come to an end.
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"We are finding it much more difficult than normal to score goals and suffering defensively too.
"We all have to improve and I'm the first one in that regard."
Atletico will be joined by local rivals Real Madrid, Manchester City and Bayern Munich in Friday's semi-final draw.
Roared on by a vociferous 55,000-strong crowd at the Vicente Calderon, Atletico were worthy winners as Barca's star front three of Lionel Messi, Luis Suarez and Neymar were limited to precious few clear-cut chances.
minutes saw him scurrying deep into his own half to dispossess Yannick Carrasco in a sign of the five-time World Player of the Year's desperation to get into the game.
Atletico made the all-important breakthrough nine minutes before half-time when Griezmann's header was this time precise and powerful enough to beat Ter Stegen from Saul Niguez's brilliant cross.
- Late controversy -
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Ter Stegen ensured Barca went in at the break just one goal down as he held a Carrasco drive at the second attempt before Niguez could follow in for the rebound.
And then as Barca struggled to clear a corner, Niguez's looping header came back off the crossbar with Ter Stegen stranded.
Griezmann had another chance to give Atletico a cushion but fired straight at Ter Stegen on his weaker right foot at the end of a lightning counter-attack.
However, Barca began to impose themselves in the final half-hour as Atletico tired and Suarez tested Oblak with a fierce effort from inside the area that the Slovenian handled impressively.
Neymar was also booked for kicking out at Juanfran moments later as Barca began to lose their cool.
Iniesta was also lucky not to be sent-off for denying a clear goalscoring opportunity as he blocked Filipe Luis's pass into Griezmann's path for the penalty.
The French international kept his head to slot home the resulting spot-kick.
However, there was still time for late controversy as Rizzoli only awarded Barca a free-kick on the edge of the box when Gabi handled.