Sharapova waited out a rain delay before starting her match against Eugenie Bouchard and then quickly dispatched Bouchard 6-2, 6-0.
That set up a third-round meeting with Elena Vesnina, who was a 6-4, 6-2 winner over Donna Vekic.
Davis lost a 3-hour match earlier this week in qualifying to 16-year-old Mallory Burdette, who then fell to Magadelena Rybarikova in the first round of the tournament.
Davis, however, became the first on the list to join the main draw if there was a withdrawal and No. 2 seed Victoria Azarenka did pull out of the event at the last minute with an injured ankle.
Not only did Davis move into Azarenka's spot, she received a bye past the first round. That sent her up against Madison Keys and Davis prevailed Friday in a match that went 2 hours, 43 minutes.
Davis will now meet Alize Cornet, who defeated Laura Robson 5-7, 7-5, 6-1.