Serena Williams, seeded ninth, 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 and set up a break point for Williams, who pulled ahead to lead 4-2.
Williams' ninth ace of the match provided a set point for the ninth-ranked American to take the first set and wrap up the match against the 68th-ranked Russian in just over an hour.
"I'm feeling good but I wasn't thrilled," said Williams, who will face Anastasia Pavlyuchenkova of Russia. "This is my first tournament in Europe so hopefully I will just keep my good form and just keep going," she added.
Venus Williams' third tournament since being sidelined by illness is over after being broken six times by the 12th-seeded German. The seven-time major champion reached the quarterfinals at Miami and Charleston, S.C., earlier this season.
Kvitova needed over an hour to beat Erakovic, with the Czech player saving all three break points against her while converting four.
Local favorite Carla Suarez Navarro rallied to beat 15th-seeded Jelena Jankovic 4-6, 7-6 (5), 6-4 to reach the second round.
Almagro saved four set points before hitting long to lose the first set in a tiebreaker. The thirteenth-ranked Spanish player broke his opponent in the opening game of the second after adjusting his game to the new court surface to win the match in three sets.
"The court isn't very good, it's very slippery," Almagro said of the unorthodox blue clay court that has been criticized by players after it was introduced for the first time in this year's tournament.