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Andreeva broke Putintseva’s serve three times in the first set and again for a 2-1 lead in the second before a brief rain delay.
After the players returned, Putintseva briefly rallied to lead 3-2, but Andreeva won the next four games to cruise into the fourth round, where she will face good friend Daria Kasatkina.
Kasatkina, competing in her first Grand Slam since switching allegiance to Australia from Russia, beat Spanish 10th seed Paula Badosa 6-1 7-5.
Third seed Jessica Pegula battled back to beat former Wimbledon champion Marketa Vondrousova 3-6 6-4 6-2 in their third-round encounter.
The American, who missed the clay-court Grand Slam last year, is bidding to reach the quarter-finals in Paris for a second time.
Pegula will face France’s Lois Boisson, who is ranked 361st in the world, after her 6-3 0-6 7-5 win over compatriot Elsa Jacquemot.
