The result left Didier Deschamps' side and Ukraine level with Finland and Bosnia in Group D after they drew 2-2 in the other game.
Les Bleus next face Kazakhstan and Bosnia away, on Sunday and Wednesday.
"Of course I can't be satisfied. We had chances to increase our advantage and we didn't take them," Deschamps said.
"But let's be honest after the break we did not exactly threaten them."
France dominated possession and they went ahead on 19 minutes as Griezmann whipped a delightful shot into the far top corner from the edge of the box.
It put Griezmann up to joint fourth with David Trezeguet in the list of France's all-time top scorers with 34 goals. Thierry Henry is top with 51.
Ukraine, however, hit back when Sergiy Sydorchuk's harmless attempt was deflected into his own net by Kimpembe 12 minutes into the second half.
"With more freshness and some momentum we could do a lot better, but a lot of my players had played in their domestic leagues on Sunday," Deschamps said.
Pogba brought steel to the midfield but Andriy Shevchenko's side were happy to sit back and wait.
Griezmann had a chance to give France a last-gap win, but his curled attempt was tipped away by Heorhiy Bushchan eight minutes from time.
Kimpembe had an opportunity to make up for his own goal in the last minute but headed the ball straight into Bushchan's arms.