Cirstea beat Caroline Wozniacki and Jelena Jankovic on her way to the last eight at Roland Garros last year.
Italian 14th seed Flavia Pennetta defeated Romania's Sorona Cirstea 6-1 1-6 6-1 to earn a second-round clash with German Andrea Petkovic.
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.
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.
Cirstea, world ranked 34th, just missed out on a seeding -- and had the Russian in trouble in the first set when taking a 3-1 advantage.
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.
But the sixth seed -- who had won just one clay-court match this year prior to facing Cirstea -- suddenly found her form in devastating fashion, winning 11 of the next 12 games to run out a comfortable 6-3 6-1 winner.
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