Anthony Martial sent a low strike wide of the post as France had their first serious attempt in the 12th minute of action, before Manchester United teammate Paul Pogba tested the palms of Gorbunov from distance.
Belarus continued to have their moments on the counter, but France were the more threatening and Olivier Giroud should have done better from a close-range volley in the 35th minute as the visitors passed up another chance.
Giroud had a golden chance to send France ahead eight minutes into the second period when Antoine Griezmann found the Arsenal forward in an advanced position, but his effort swung wide of the post.
Griezmann came close himself with a swinging free kick just before the hour, but Gorbunov was on hand to make a smart save, before Giroud hit the crossbar with a close-range header moments later.
The 68th minute brought another opportunity for the visitors as Moussa Sissoko delivered a low cross in to Griezmann, but once again the home side's goalkeeper made the save to somehow keep the score all square.
Pogba and Griezmann then both had opportunities either side of the 70-minute mark as the onslaught continued, while Kevin Gameiro also came close after being introduced off the bench, but the visitors could not make the breakthrough.
Next up for France is a home match against Bulgaria on October 7, while Belarus will travel to Netherlands, who drew 1-1 in Sweden on Tuesday, on the same night.