Australian No.21 seed Alex de Minaur romped to a 6-1, 6-4, 6-4 victory over American Bradley Klahn, however, 15th seed Nikoloz Basilashvili fell 6-4, 6-1, 6-3 to Argentine Juan Ignacio Londero.
Another Aussie, Ashleigh Barty, cruised to a 6-3 6-3 win over American Jessica Pegula.
Pegula's compatriot, 32nd seed Frances Tiafoe, went down to Serb Filip Krajinovic, falling 6-0 in the fifth set.
Last year’s Wimbledon semi-finalist and No.18 seed Julia Goerges was another early seeded casualty. The German had split six prior meetings with two-time Roland-Garros quarter-finalist Kaia Kanepi, of Estonia, but faded to a 7-5, 6-1 defeat.
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Also on Monday, women’s No.4 seed Kiki Bertens extended her clay-court win streak to 13 matches with victory over French wild card Pauline Parmentier, while recent Rome finalist and No.26 seed Johanna Konta eased past Antonia Lottner 6-4, 6-4.