They recovered in the second half, however, and sealed the three points when Griezmann scored his 35th international goal in the 60th minute.
The result means that the Blues lead Group D with seven points, four points ahead of second-placed Ukraine.
Hugo Lloris, whose 123rd appearance put him on par with Thierry Henry as France’s second most capped player, was called up twice shortly before the half hour mark.
First, he fought off Darko Todorovic’s fierce strike from a distance before superbly knocking down Anel Ahmedhodzic’s point-blank head from a corner.
Kylian Mbappe and Griezmann cut isolated attacking numbers for France in the opening 45 minutes, with Deschamps’ side entering at the break having only had one shot on target.
They improved after the break and took the lead on time, with Griezmann directing a superb header from Adrien Rabiot’s cross into Ibrahim Sehic’s top right corner.
The hosts rarely seemed to recover a point in the final stages as France comfortably held command of the group.