Heading into stoppage time and searching for the goal that would have forced an extra 30 minutes, Barca were incensed not to receive a penalty when Andres Iniesta's shot struck the arm of Atletico captain Gabi just inside the box, with the referee instead awarding a free-kick which Lionel Messi hit over.
Atletico Madrid 2 Vs Barcelona 0
Atletico Madrid are through to the Champions League semi-finals after beating holders Barcelona 2-0 at the Vicente Calderon on Wednesday.
Luis Enrique's men had led 2-1 from the first leg but ultimately failed to cope with the intensity of an Atletico side who are also just three points behind them in La Liga and will be buoyed by this scalp as they seek to heap added misery on to their Catalan counterparts domestically.
Barcelona's willingness to sit on that slender first-leg lead led to a cagey and tense first half-hour, but Atletico broke through with 36 minutes on the clock, as Griezmann nodded in to spark wild celebrations.
Dogged Atletico held firm for the remainder of the half and went desperately close to a second soon after the break, but Lucas Hernandez's header was kept out by the crossbar.
Barca soon began to exert some control, keeping Atletico penned back, but Jan Oblak had little trouble keeping Luis Suarez at bay with their best chance just after the hour mark.
Atletico were forced to withstand significant pressure during the final stages, but Diego Simeone's men held on in typical fashion and then netted a second on the break as Iniesta's handball allowed Griezmann to convert from the spot with two minutes remaining.
Referee Nicola Rizzoli then provoked a furious reaction from Barca with his decision to award only a free-kick for Gabi's handball, and the result means Atletico have knocked Barca out in the quarter-finals of this competition in two of the past three seasons, while they gain further psychological advantage in their quest to chase down Barca's slender lead in the league with just six games to go.
Urged on by a raucous atmosphere at the Vicente Calderon, Atletico began proceedings with great urgency, pouring men forward quickly and in numbers.
It took just seven minutes for the hosts to create the first chance and they ultimately let Barca off the hook, as the unmarked Griezmann headed straight at Marc-Andre ter Stegen.
Barca's terrifying attack of Messi, Neymar and Suarez were being kept quiet, but the away side did get to grips with Atletico's intense pressing, resulting in the two sides cancelling each other out in midfield for long periods.
Atletico's patience and intensity paid off nine minutes before the break, though, as Griezmann rose to meet Saul Niguez’s delightful cross and sent a fine glancing header past the helpless Ter Stegen.
Simeone's men could have added a second just before the interval, but Yannick Ferreira Carrasco was thwarted by Ter Stegen after single-handedly leading a breakaway.
Barca initially struggled to mount a response after the break and Atletico went mightily close to increasing their lead, as Lucas' looping header bounced back off the bar in the 54th minute.
A crucial block from Gabi six minutes later prevented Suarez a simple tap-in, but the occasion marked an improvement from the visitors.
Oblak was their saviour in the 66th minute, as the goalkeeper got himself in the way of Suarez's effort after the Uruguayan expertly turned his compatriot Diego Godin on the edge of the area.
Wave after wave of Barca attacks were halted by the home defence during the final 20 minutes, as the reigning champions continued to be frustrated, with Messi kept particularly quiet.
Barca threw everything at Atletico in desperate fashion for the final 10 minutes, but it was the home side who were left jubilant.
A marvellous Filipe Luis breakaway left Iniesta as the last man and the midfielder handled in the penalty area, with Griezmann squeezing the spot-kick just beyond Ter Stegen diving low to his left.
Barca should have been presented with the chance to force extra-time, but luck was on Atletico's side as the holders were dumped out.