Griezmann headed home Saul Niguez's exquisite cross to give the home side the lead in the 36th minute and provide them with a platform from which they comfortably contained their uncharacteristically limp opponents to seal a 3-2 aggregate win.
The France forward scored again from the penalty spot with two minutes remaining, following a handball from Andres Iniesta, to complete a 3-2 aggregate victory that eliminated Barca at the quarter-final stage for the second time in three seasons. Barcelona, who were bidding to become the first team to retain the Champions League, seem to have run out of steam at the end of the season, having won just one of their last five matches.
They have not looked the same side since their 39-game unbeaten run ended at the start of the month and they came into the Wednesday's match on the back of a 1-0 defeat at Real Sociedad.
"I congratulate Atletico Madrid, they were better than us,” Barca coach Luis Enrique told reporters.
“We had control without creating much danger. The team wanted to retain the title, the fans did too, but today it was not to be. We are sad, of course.”
The Atletico supporters created a fervent atmosphere and the noise at kick-off spurred the hosts into a strong start, with Gabi blazing over and Yannick Carrasco and Griezmann testing Barca keeper Marc-Andre ter Stegen. Atletico took the lead in the tie on away goals when Saul carved out a cross with the outside of his boot towards Griezmann who headed into the top corner.
"We never stopped believing, we never give up what we want to do,” Atletico coach Diego Simeone said.
"It was lovely to watch from the touchline. These moments won’t come back because they are unique in football. After Griezmann doubled the lead from the spot, Barcelona still had two minutes plus stoppage time to find a goal that would have forced extra time and Lionel Messi nearly conjured a late leveller but sent a free kick over the bar.
“We were playing against one of the best teams in the world and we did ourselves proud, now we’re in the semi finals, among the top four teams in Europe,” Griezmann said.
“Our fans support us in every home game and push us to produce performances like tonight's. This game belongs to them too."
(Editing by Ed Osmond)
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