Halep was a 6-4 6-4 winner against Alize Cornet, who defeated her in the fourth round of the Australian Open in January, and will face seventh seed Danielle Collins next.
It was only the Romanian's second win in her six matches against Cornet, though she needed two chances to serve out both sets.
Azarenka needed just 68 minutes to see of Viktorija Golubic, dropping just five points behind her first serve and winning 12 out of 14 from the final three games to complete a 6-3 6-0 triumph.
It only took Gauff two minutes more to see off the challenge of Kerber, who has now won just two of her eight matches on the WTA Tour this year.
Leylah Fernandez came from behind to claim an impressive 1-6 6-2 6-2 victory over 16th seed Anastasia Pavlyuchenkova, while Madison Keys went down 6-4 6-4 to Anhelina Kalinina.
Aliaksandra Sasnovich's 3-6 7-6 (7-4) 6-1 win over Veronika Kudermetova secured her a second-round match against second seed Paula Badosa.
Belinda Bencic overcame Elisabetta Cocciaretto 6-4 6-2, Shuai Zhang set up a match with Aryna Sabalenka by beating Martina Trevisan 6-4 6-2, and Petra Martic will face Anett Kontaveit after besting Karolina Muchova 6-2 3-6 6-1.
There were also wins for Daria Kasatkina and Yulia Putintseva before rain interrupted Ajla Tomljanovic's match against Camila Giorgi when the Australian was 6-3 4-2 up.