Griezmann struck in each half after his partnership with Karim Benzema bore fruit in satisfying fashion to put the world champions on 12 points from six matches at the top of the group.
Finland, who had a decent opening half, are third on five points but have two games in hand of the French.
Bosnia are fourth on three points from four games after a 2-2 draw with bottom side Kazakhstan, who also have three points. Ukraine are second on five points from as many matches.
"Even it was not perfect we showed determination and with that came some more technical quality," said France coach Didier Deschamps.
"We had more confidence, it was more like us. It's a very important result for us, now we have to finish the job in November."
After two 1-1 draws against Bosnia and Ukraine, Deschamps' tinkering paid off as France started in an usual 5-2-3 formation with Leo Dubois and Theo Hernandez as fullbacks and Griezmann, Anthony Martial and Benzema up front.
But it was Finland who had the first clear chance through Urho Nissila, whose floating strike was tipped over the bar by Hugo Lloris.
Les Bleus, however, were on the attack and Lukas Hradecky pulled off a nice save in the 22nd minute to deny Benzema.
Griezmann found the back of the net when he whipped the ball past Hradecky with the outside of his foot after collecting a clever deflection in the box by Benzema on 25 minutes.
Finland continued to threaten on the break, but France proved clinical early in the second half.
After some more nice work from Benzema, Dubois found Griezmann in the area and the forward sneaked the ball between Hradecky and the left-hand post to give the hosts some breathing space in the 53rd.
He has now scored 41 international goals to become France's joint third all-time top scorer alongside Michel Platini.
Shortly afterwards, Martial missed Hernandez's cross for what would have been France's third.
France controlled the end game, keeping the ball high on the pitch and creating a few chances through Benzema, who had a night to remember on his visit to his former stadium, 12 years after leaving Lyon for Real Madrid.