The visitors were on the back foot for almost the entire game after defender Sime Vrsaljko was dismissed for two yellow cards in the first 10 minutes.
With Atletico’s furious manager Diego Simeone sent to the stands shortly afterwards, the Spanish side survived unscathed to half-time, with Alexandre Lacazette clipping the post with a half-volley the hosts’ best chance.
Lacazette nodded in Jack Wilshere’s cross just after the hour to apparently edge the Gunners closer to a Europa League final send-off for outgoing manager Arsene Wenger.
But with eight minutes remaining, Griezmann latched on to a long ball forward, shrugged off Laurent Koscielny and stabbed into the net at the second attempt for an away goal to swing the tie Atletico’s way going into the return leg in Madrid next Thursday.
Arsenal’s fans were left ruing a familiar combination of wasteful finishing and costly lapses in concentration as the teams departed to a muted response in Wenger’s final home European game.
Their team now must breach an Atletico defence that has conceded just four times at home in La Liga this season to have any hope of extending Wenger’s 22-season spell in charge to the final on 16 May.
In contrast to Arsenal, Atletico Madrid’s defensive discipline has served them well in recent continental campaigns, grinding out victory in four of their five European semi-finals over the past nine seasons.
But the Spaniards’ usual pragmatism deserted them as they self-combusted in the opening 13 minutes.
Croatian right-back Vrsaljko was booked as he caught Wilshere late in the second minute and, instead of showing the requisite caution, blundered into Lacazette with his studs up eight minutes later.
So early in the contest, Atletico may have expected referee Clement Turpin to limit punishment to a stern and final warning.
But the 35-year-old Frenchman judged the challenge rather than the occasion and showed Vrsaljko a justified second yellow.
Simeone was exasperated by that decision and incensed by Turpin opting not to book Hector Bellerin shortly after. The Argentine was dispatched to the stands after aiming a stream of invective at the official.