Daria Gavrilova was among the seeded players to advance from the opening round of the WTA tournament in South Carolina.
The 11th-seeded Australian dropped the opening set before she overcame Ons Jabeur of Tunisia 1-6 6-4 6-3.
Gavrilova will meet Camila Giorgi in round two after the Italian eased past Silvia Soler-Espinosa 61- 6-4.
Australian ninth seed Ashleigh Barty and countrywoman Samantha Stosur commence their campaigns on Tuesday.
Other seeded players to progress on Monday included Naomi Osaka, the Japanese 10th seed, who beat Jennifer Brady 6-4 6-4.
Dutch 12th seed Kiki Bertens of Netherlands, 14th-seeded Frenchwoman Alize Cornet of France and 15th-seeded Mihaela Buzarnescu all won through.
Other opening-round winners were Christina McHale, Aleksandra Krunic, Bernarda Pera, Laura Siegemund, Varvara Lepchenko, Fanny Stollar, Mihaela Buzarnescu, Caroline Dolehide and Polona Hercog.
