Romania's Sorana Cirstea earned a clash with former world No. 1 Jelena Jankovic after eliminating German sixth seed Angelique Kerber 6-4 6-0.
In the accompanying WTA tournament at the same venue, Russia's Maria Sharapova began with a 6-2 6-2 win over Romania's Sorana Cirstea.
In the women's section, Serena Williams thrashed Caroline Garcia 6-1 6-2 to set up a last-16 clash with Romania's 26th seed Sorana Cristea.
Elsewhere, second seed Venus Williams faces Swiss veteran Patty Schnyder, while defending champion Svetlana Kuznetsova of Russia opens her campaign against Romania's Sorana Cirstea.
Holder Svetlana Kuznetsova overcame a shaky start to reach the second round of the French Open with a straight sets victory over Romania's Sorana Cirstea on Sunday.
Serena and Venus also won their opening doubles match, beating Romania's Sorana Cirstea and Simona Halep 6-3 6-2 to set up a clash with Kerber and Sabine Lisicki.
Chakvetadze will take on Polish eighth seed Agnieszka Radwanska in the last 16, with the winner to then play either German 17th seed Sabine Lisicki or unseeded Romanian Sorana Cirstea.
In her last five tournaments, though, she has reached at least the quarterfinals, and she started on an up note in Paris Monday as she drubbed Sorana Cirstea 6-2, 6-1.
Tenth seed Shahar Peer completed a routine 6-3 6-2 victory over Romania's Sorana Cirstea to secure a match against Italy's 22nd seed Flavia Pennetta , who beat Spain's Lourdes Dominguez Lino 6-2 6-2.
Montenegro, a country whose economy relies heavily on tourism and has been developing its status as a boutique resort for wealthy Russians, has had healthy relations with Russia, thanks in part to its relatively peaceful stance during the Balkan wars, according to Sorana Parvulescu of Control Risks Group, a specialist risk consultancy.
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