A two-time runner-up in Monterrey, top seed Angelique Kerber moved into the second round.
While most of the stars advanced at the Miami Open, Elina Svitolina and Garbine Muguruza were beaten.
Bianca Andreescu's impressive start to the year continued as she set up an Indian Wells Open semi-final against Elina Svitolina.
Simona Halep and Elina Svitolina were in good form at the Indian Wells Open on Sunday.
Garbine Muguruza is into the Indian Wells Open last 16 after Serena Williams retired early into the second set of their third-round match.
Untroubled by Vera Lapko, Victoria Azarenka next faces Serena Williams at the Indian Wells Open.
While Elina Svitolina, Simona Halep, Petra Kvitova and Angelique Kerber all won on Tuesday, Naomi Osaka suffered an upset.
Day one of the Dubai Tennis Championships saw the likes of Garbine Muguruza and Daria Kasatkina advance to the second round.
World number two Petra Kvitova's title defence at the St Petersburg Ladies Trophy ended at the quarter-final stage on Friday.
Garbine Muguruza trailed 4-0 in the second set, but won the next five games as Mona Barthel was unable to force a decider.