Williams had 13 aces and 40 winners against Vesnina, who was struggling with her game on the slippery blue clay court. The Russian lost her balance in the sixth game to set up a break point for Williams, who pulled ahead to lead 4-2.
Williams' ninth ace of the match earned her a set point. The ninth-ranked American wrapped up the match against the 68th-ranked Russian in just over an hour.
''I'm feeling good but I wasn't thrilled,'' Williams said. ''This is my first tournament in Europe so hopefully I will just keep my good form and just keep going.''
In the men's tournament, Spain's Nicolas Almagro rallied to beat Santiago Giraldo of Colombia 6-7 (5), 6-3, 6-3.
American Ryan Harrison beat Sergiy Stakhovsky of Ukraine, 7-6 (3), 7-6 (5) to reach the second round.
Other winners Monday included Serbia's Victor Troicki and Mikhail Youzhny of Russia.