Didier Deschamps' men scored the opening goal for the first time in six matches, Griezmann delivering a stunning left-footed finish to become the fourth player to score 40 goals for France.
Finland were the most recent defeat team to triumph on French soil but rarely troubled in Lyon, Griezmann adding a second after the break to draw him level with Michel Platini in third on the all-time scoring list.
The host sits top of Group D and remain unbeaten despite its underwhelming return after a poor Euro 2020, while Markku Kanerva's men have won just two of its past 12 games across all competitions.
Urho Nissila brought the first save from Hugo Lloris, his looping strike forcing the Tottenham goalkeeper to tip over before Karim Benzema called Lukas Hradecky into action with a left-footed effort.
However, Hradecky was no match four minutes later as Griezmann beautifully curled into the bottom-right corner after an offload from Benzema, who was appearing in Lyon for the first time since 2011 after returning to his former club with Real Madrid.
Paul Pogba came close to adding a second, though he could only volley wide on the stretch as he sliced his right-footed strike from outside the area.
Griezmann doubled his and France's tally after the break, rolling a delicate shot through the legs of Hradecky after being teed up by Leo Dubois.
Atletico Madrid's new loanee should have completed his hat-trick, Benzema again acting as the creator, but he could only poke over before acrobatically volleying wide with his next attempt.
Benzema could have had a goal of his own in the closing stages, but Hradecky got down well to parry away after the forward had struck a left-footed volley.