Devvarman, who is ranked 315 in the ATP list, needed just over an hour to dispose off the Russian's challenge with a 6-1, 6-4 scoreline.
Devvarman put up a solid performance against the Russian, who is 148 places above the Indian, ranked 167 in the list.
India's best singles player at the moment broke his Russian opponent's serve four times to take the first set at 6-1.
In the second set, both Devvarman and Kunitsyn held their serves during the first six games to be level at 3-3.
The Indian then held his serve in the next game to go 5-3 up. Kunitsyn however did well as he held onto his serve.
In the 10th and final game, Devvarman won on his second match point as forehand from the Russian went over the baseline.