The 32nd-ranked Benneteau, who made the finals at the Rotterdam event this year, prevailed over hard-serving South African Anderson 7-6 (7-4), 6-2 at Devonshire Park.
Australian Bernard Tomic rebounded from a recent dry spell with a 6-3, 6-4 decision over British wild card James Ward, while Czech veteran Radek Stepanek dispatched Marinko Matosevic of Australia 6-3, 6-2.
Other opening-round winners Tuesday included Croat Ivan Dodig, Slovak Martin Klizan and Spaniard Feliciano Lopez.
Notable deaths of 2014 [PHOTOS]
UPI Sports Results for September 19, 2014