Atletico, who have reached at least the quarter-finals four years in in succession, were heading out of the competition before the forward's glorious impact after 69 minutes.
Griezmann lit up a tense Wanda Metropolitano with a wonderful scissor-kick volley to break the deadlock before setting up fellow Frenchman Kevin Gameiro with a perfect pass to clinch the points after Bruno Peres was dismissed.
Even though he had not scored in eight Atletico appearances in a row, his worst run for the club, Griezmann claimed he had not been worried about his problems in front of goal.
"It didn't come into it," Griezmann told beIN SPORTS when asked if he had been anxious about his form.
"I tried to do the best I could in training and for my national team.
"I'm happy with the goal, for my team-mates, for being able to help them. Hopefully we can improve, me first and foremost. I try to give my very best. We have to keep working."
Atletico, semi-finalists last season, must win at already qualified Chelsea on matchday six to have any hope of qualifying from Group C, although Roma - two points clear of Diego Simeone's side - are in pole position as they face bottom side Qarabag.
But Griezmann said Atletico will keep fighting to stay in the Champions League despite their chances of qualification looking slim.
"We're going to try until the end," Griezmann said. "We knew it was important to win. We'll see what happens now.
"[Chelsea will be] a great game. We're going there to win."
Roma defender Federico Fazio accepted his side cannot rely on Atletico's result at Stamford Bridge next month after Roma's six-game winning run came to an end in the Spanish capital.
"We played a decent first half with some good possession, but we lost the ball more in the second half and this helped them," Fazio told Mediaset Premium .
"We worked hard at the back and we've got to carry on like this, keeping the faith. Now we desperately want to beat Qarabag and I'm sure our fans will help us."