The German, a surprise absentee from the main show courts due to a packed schedule, took the opening set 6-2.
But Kerber won just three more games as American Davis, beaten in qualifying but handed a second chance in the main draw, won 2-6 6-2 6-1 to knock out the fifth seed.
World number one Ashleigh Barty suffered no such trouble on the same court, the Australian sweeping past Alison Van Uytvanck 6-1 6-3.
Barty will face Harriet Dart, who was joined in round three by fellow Brit Johanna Konta, a straight-sets winner against Katerina Siniakova.
Sloane Stephens awaits British number one Konta, the American ninth seed dropping just two games against China's Wang Yafan.
Two-time champion Petra Kvitova is through after a straight-sets win over Kristina Mladenovic of France.
The Czech sixth seed triumphed 7-5 6-2 in an hour and 25 minutes on Court One.
Fourth seed Kiki Bertens came from a set down to beat American Taylor Townsend 3-6 7-6 (5) 6-2.