Griezmann joined Atletico's top 10 scorers in European competition by putting the home side ahead in the 28th minute with an improvised volley.
Sardar Azmoun levelled barely two minutes later, becoming the first player to score against Diego Simeone's side in the competition this season.
Atletico's passage to the last 16 was guaranteed nonetheless by Bayern Munich's 2-1 win at PSV Eindhoven but Griezmann beat the offside trap in the dying seconds to ensure last year's runners-up held on to their perfect record.
Atletico top Group D on 12 points, Bayern are second on nine, while Rostov and PSV, with a point each, will compete for third place and a spot in the Europa League.
Simeone gave a debut to Croatian defender Sime Vrsaljko but otherwise opted for a familiar starting 11 in the return game with Rostov after snatching a 1-0 win in Russia a fortnight ago.
Rostov's five-man defence managed to frustrate Atletico for the majority of the game and the Spanish side's opener came from a moment of brilliance from Griezmann, who was on a run of five games without a goal for club or country going into the fixture.
Yannick Carrasco collected the ball from Filipe Luis after a corner looked to have gone to waste and his chipped pass found its way to Griezmann via a weak header from Rostov's Azmoun. The Frenchman, with his back to goal, reacted quickly to hook the ball in on the volley.
Azmoun promptly made amends at the other end though, receiving a pass from Dmitry Poloz as Atletico were caught on the break and slipping the ball past Jan Oblak.
Atletico laid siege to the Rostov goal after the break but were repeatedly frustrated by goalkeeper Soslan Dzhanayev, who turned away efforts from Griezmann, Diego Godin, Carrasco and Kevin Gameiro.
Griezmann initially did not celebrate his last-gasp winner, believing he was offside, but peeled away in joy as the linesman kept his flag down, correctly ruling that the previous touch had come from a Rostov player.
(Reporting by Richard Martin; Editing by Clare Fallon)
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