Roger-Vasselin ground out a 3-6, 7-6 (5), 7-5 victory over big-serving Ante Pavic of Croatia, before Dodig followed up with a 7-5, 6-2 win against wild card Olivetti of France.
Pavic had 23 aces, compared to 13 for Roger-Vasselin, who saved seven of the nine break-point chances he faced. He next plays Ukraine's Sergiy Stakhovsky after he rallied past qualifier Daniel Evans 3-6, 7-6 (4), 6-4.
Dodig, who saved all 11 break-points he faced and broke Olivetti three times, will play Germany's Tobias Kamke in the second round after he beat Britain's Kyle Edmund 7-5, 6-1.
Davydenko, who reached a career highest ranking of No. 3, next plays second-seeded Jo-Wilfried Tsonga, the defending champion. Tsonga leads 3-2 in their career head-to-heads.