The France striker opened the scoring against Rostov with a brilliant flick, but Sardar Azmoun equalised - only the fourth time in 19 home Champions League games under Diego Simeone that Atletico have conceded - before Griezmann fired in off the crossbar in injury time.
Bayern won 2-1 at PSV, with Poland striker Robert Lewandowski scoring twice and hitting the woodwork three times - taking his tally for club and country this season to 19 goals in 17 games.
Paris St-Germain also went through thanks to a 2-1 victory at Basel.
Thomas Meunier scored a brilliant late volley for the French champions, who qualified with Arsenal from Group A.
A draw would have been sufficient for Unai Emery's PSG side thanks to Arsenal's 3-2 win at Ludogorets.
Napoli missed the opportunity to clinch their last-16 place as they drew 1-1 at Besiktas.
Ricardo Quaresma gave the Turkish champions a 78th-minute lead before Marek Hamsik curled home a fantastic 25-yard effort four minutes later.
Benfica beat Dynamo Kiev 1-0 courtesy of Eduardo Salvio's 45th-minute penalty.
Borussia Dortmund will progress if they beat Sporting Lisbon at home. Real Madrid will join them from Group F if they win at Legia Warsaw, providing Dortmund win.
Leicester will go through if they beat FC Copenhagen at the King Power Stadium. A draw would be enough for the Foxes if FC Porto do not beat Club Brugge.
Victory over Lyon will secure Juventus' place in the last 16, which would mean Sevilla will qualify if they beat Dinamo Zagreb.
Match ends, Atlético de Madrid 2, FC Rostov 1.
Second Half ends, Atlético de Madrid 2, FC Rostov 1.
Attempt missed. Christian Noboa (FC Rostov) right footed shot from outside the box is too high from a direct free kick.
Foul by Koke (Atlético de Madrid).
Moussa Doumbia (FC Rostov) wins a free kick in the attacking half.
Goal!  Atlético de Madrid 2, FC Rostov 1. Antoine Griezmann (Atlético de Madrid) left footed shot from very close range to the top right corner.
Offside, Atlético de Madrid. Gabi tries a through ball, but Diego Godín is caught offside.
Attempt missed. Sime Vrsaljko (Atlético de Madrid) right footed shot from long range on the right misses to the left. Assisted by Koke.
Attempt saved. Stefan Savic (Atlético de Madrid) header from the centre of the box is saved in the bottom left corner. Assisted by Koke with a cross.
Kevin Gameiro (Atlético de Madrid) wins a free kick in the attacking half.
Foul by Aleksandru Gatcan (FC Rostov).
Foul by Ángel Correa (Atlético de Madrid).
Andrei Prepelita (FC Rostov) wins a free kick in the defensive half.
Attempt blocked. Ángel Correa (Atlético de Madrid) left footed shot from the centre of the box is blocked.
Foul by Gabi (Atlético de Madrid).
Christian Noboa (FC Rostov) wins a free kick in the defensive half.
Attempt missed. Christian Noboa (FC Rostov) left footed shot from outside the box is too high. Assisted by Dmitriy Poloz following a set piece situation.
Filipe Luis (Atlético de Madrid) is shown the yellow card for a bad foul.
Foul by Filipe Luis (Atlético de Madrid).
Dmitriy Poloz (FC Rostov) wins a free kick on the right wing.
Soslan Dzhanaev (FC Rostov) is shown the yellow card.
Attempt saved. Kevin Gameiro (Atlético de Madrid) right footed shot from outside the box is saved in the centre of the goal. Assisted by Fernando Torres.
Corner,  Atlético de Madrid. Conceded by César Navas.
Substitution, Atlético de Madrid. Ángel Correa replaces Yannick Carrasco.
Attempt saved. Antoine Griezmann (Atlético de Madrid) left footed shot from outside the box is saved in the top centre of the goal. Assisted by Gabi.
Substitution, FC Rostov. Denis Terentjev replaces Timofei Kalachev because of an injury.
Attempt blocked. Antoine Griezmann (Atlético de Madrid) header from the centre of the box is blocked. Assisted by Sime Vrsaljko with a cross.
Substitution, FC Rostov. Moussa Doumbia replaces Sardar Azmoun.
Aleksandru Gatcan (FC Rostov) is shown the yellow card for a bad foul.
Fernando Torres (Atlético de Madrid) wins a free kick in the attacking half.
Foul by Aleksandru Gatcan (FC Rostov).
Attempt saved. Fernando Torres (Atlético de Madrid) right footed shot from the right side of the box is saved in the centre of the goal. Assisted by Kevin Gameiro.
Attempt missed. Yannick Carrasco (Atlético de Madrid) left footed shot from outside the box misses to the right following a corner.
Corner,  Atlético de Madrid. Conceded by Vladimir Granat.
Attempt blocked. Antoine Griezmann (Atlético de Madrid) right footed shot from outside the box is blocked. Assisted by Yannick Carrasco.
Attempt missed. Koke (Atlético de Madrid) right footed shot from outside the box misses to the right. Assisted by Filipe Luis following a corner.
Corner,  Atlético de Madrid. Conceded by Sardar Azmoun.
Kevin Gameiro (Atlético de Madrid) wins a free kick on the left wing.
Foul by Timofei Kalachev (FC Rostov).
Attempt missed. Fernando Torres (Atlético de Madrid) left footed shot from the left side of the six yard box is close, but misses to the left. Assisted by Kevin Gameiro with a cross.
Antoine Griezmann scored a last-second winner as Atletico Madrid qualified for the Champions League last 16 alongside Group D rivals Bayern Munich.