Griezmann had gone eight games with a goal for Atleti - his worst run for the club - but broke the deadlock after 69 tense minutes at the Wanda Metropolitano on Wednesday.
Angel Correa did superbly to keep the ball in at the byline and his cross was hammered in by Griezmann's stupendous scissor-kick volley.
After Bruno Peres saw red for a second yellow card, Griezmann then slipped Kevin Gameiro clear to add a second and ensure Atletico will travel to already qualified Chelsea on matchday six with some hope of qualification.
Atletico have struggled for goals of late and the defeat of Roma was the first time in 11 matches in all competitions that Simeone's side have scored more than once.
"This team has a lot of heart. We had a responsibility to win and we managed it," Simeone told beIN SPORTS.
"It was an extraordinary goal, with a good attacking involvement of the whole team. Hopefully this is the beginning of an alliance with the goal!
"We have a obligation to beat Chelsea in London but Roma also have Qarabag on their own pitch.
"We have allowed ourselves the possibility of continuing to have the chance to progress. It's down to us first, winning our match - and then fate will take us where it will."
4 - Each of the last four Roma's defeats in the Champions League (excl. Qualifiers) has come against Spanish sides. Allergy. #AtleticoRoma— OptaPaolo (@OptaPaolo) November 22, 2017
Although Roma's six-game winning streak in all competitions ended at the Wanda Metropolitano, coach Eusebio Di Francesco claimed the Serie A side can learn from their performance.
"Euphoria is good, but arrogance isn't. We need to realise that," Di Francesco said. "We're going through a process of development and it's better if you don't lose, but you need to be intelligent about your defeats and work on your mistakes.
"Given the team we have, we made too many technical mistakes tonight, but we can't forget that Atletico cause problems for everyone they face.
"It was very even. We've shown we can go toe to toe with anyone and proved we have bags of character.
"Let's not forget who we've been up against tonight. They're all quality players. We matched them until they scored, but we lacked precision when we won the ball in the first half - there were at least 11 or 12 times we lost the ball in their half of the field.
"We needed to be more decisive with the final ball and go in search of the goal. They didn't create very much at all in the first 45 minutes, but after the break they showed more hunger and desire."