The fourth-seeded Anderson won 6-3, 6-4, 6-2 in one hour and 45 minutes.
The 2.03m tall South African hit 16 aces and never lost his serve, while breaking the Frenchman four times.
Anderson, last year’s finalist, will next face either Serbia’s Janko Tipsarevic of Japan’s Yoshihito Nishioka.
Meanwhile, South Africa’s other participant in the men’s singles, Lloyd Harris , will face the second-seeded Roger Federer on Tuesday.
In the men’s doubles, South Africa’s Raven Klaasen and his New Zealand partner Michael Venus will open their account against British duo Jonny O’Mara and Luke Bambridge on Wednesday.
Klaasen and Venus, who were also runners-up last year, are the third seeds.