Azarenka easily beat Alize Cornet of France 6-1, 6-0, while Sharapova had a harder time knocking off Simona Halep of Romania 7-5, 7-5.
Azarenka is hoping to rebound from last week in Tokyo where she withdrew from the tournament after a bout of dizziness in the third round.
Azarenka has won four times this year, including the Australian Open. Sharapova won her first Grand Slam title in four years at the French Open.
Caroline Wozniacki of Denmark, seeded 10th, also advanced, beating Chanelle Scheepers of South Africa 7-5, 6-7 (8), 6-2. And Lara Arruabarrena-Veinco of Spain defeated Zheng Jie of China 6-2, 6-4.
Other first round winners included Sorana Cirstea of Romania, who downed Sofia Arvidsson of Sweden 6-2, 6-3, and Carla Suarez Navarro of Spain, who beat fourth-seeded Petra Kvitova of the Czech Republic.
In the first round of the men's draw, fourth-seeded Marin Cilic of Croatia, last year's runner-up, lost to Marius Copil of Romania 3-6, 6-7 (0), 6-4.
In other men's matches, Florian Mayer of Germany beat Fabio Fognini of Italy 6-1, 6-2, while Carlos Berlocq of Argentina defeated Marinko Matosevic of Austria 3-6, 6-2, 6-3.
Sixth-seeded Alexandr Dolgopolov of Ukraine beat Guillermo Garcia-Lopez of Spain 6-0, 6-4, and Sam Querrey of the United States defeated Fernando Verdasco of Spain 5-7, 6-3, 6-4.