Griezmann, who rejoined Atletico last month on a season-long loan from Barcelona, pounced in the 84th minute to level, before Suarez grabbed Atletico their first win in this season's competition in the 97th minute.
After a lightning start in a frenetic atmosphere, Milan raced into a 20th-minute lead through Rafael Leao, but the home fans were silenced nine minutes later as Franck Kessie was sent off after picking up two yellow cards in quick succession.
As the game wore on, Milan retreated further and further back and Griezmann finally made the Atletico pressure count, firing into the bottom corner to level.
"I'm very proud to be here," Griezmann said. "I think the fans want to see me like this like tonight. I want to bring them happiness and they'll be going to bed happy tonight."
Pierre Kalulu was then adjudged to have handled the ball in stoppage time to give Atletico the chance to snatch a dramatic victory at the death, with Suarez converting from the spot to a chorus of boos at the Siro Siro.
"I don't think it was the best evening for the referee," Milan coach Stefano Pioli said. "The numerical inferiority forced us to play another type of match, after the sending off there were many difficulties."
Milan supporters had been waiting a long time for this sort of occasion. Atletico were in fact the last side to face them in the Champions League in the San Siro before Tuesday's encounter, back in February 2014, when a Diego Costa goal was enough to earn the Spaniards victory.
The iconic stadium was abuzz ahead of the match. Stadiums are currently permitted to be only 50% full in Italy, but the majority of those fans were concentrated together in a packed Curva Sud and roared out their team for the warm-up like they had just won the final.
When the Curva unveiled their banner, in English, pre-match, the message was a simple one: "Milan is back".
The raucous crowd, who bellowed out the Champions League anthem with gusto, seemed to give Milan extra energy, as they flew out of the traps early on.
Moments after Rebic should have opened the scoring, Leao did, drilling home to spark wild scenes of celebration in the stands.
That euphoria was short-lived, however, as Kessie was given his marching orders following his second booking for a foul on Marcos Llorente before the half-hour mark, much to the hosts' fury.
"I'm sure [the people at home enjoyed it] more than us. The end of the second half was really good, the bravery of the team, with an extra player, being able to attack more freely, but before that ... Milan were better than us. They had more rhythm, more dynamic, they scored," Atletico manager Diego Simeone said.
On Griezmann's performance, Simeone said: "I'm really pleased with him, a great goal from an extraordinary player like he is, we need more from him and we'll demand it, as he knows."
The result leaves Milan without a point at the bottom of Group B, while Atletico are second with four, trailing leaders Liverpool by two points following the Premier League side's 5-1 win over FC Porto in Portugal.