The German two-time runner-up and top seed Kerber and defending champion Muguruza both moved into the second round in Mexico.
Angelique Kerber and Garbiñe Muguruza booked their spots in the Monterrey Open second round on Tuesday.
Kerber, the top seed and two-time runner-up, overcame qualifier Kristina Kucova 7-6 (7-4) 6-2 on the hard-courts in Mexico.
The German's two final defeats at the WTA International event have come to four-time champion Anastasia Pavlyuchenkova.
Pavlyuchenkova started her bid for a fifth crown with a 6-0 6-3 thrashing of Russian compatriot Anna Blinkova.
Meanwhile, defending champion Muguruza was tested before getting past Romanian Elena-Gabriela Ruse 6-4 7-6 (7-4).
Also in Monterrey, two-time grand slam champion Victoria Azarenka eased through her opener against qualifier Miyu Kato 6-4 6-2.
Kristina Mladenovic battled through, while the only seed to fall was Alison Riske, who was beaten by Sachia Vickery 7-6 (8-6) 6-1.