Davis, the third seed and last year's runner-up at the WTA International event, was stunned 7-5 6-3 by Belarusian teenager Aryna Sabalenka.
The American world number 34 has enjoyed a decent 2017, but she was beaten by a player ranked 87 places below her.
There were no such problems for fellow seeds Julia Goerges and Oceane Dodin on the hard courts.
Fourth seed Goerges was too good for Francoise Abanda 6-4 6-4, while fifth seed Dodin led Jelena Jankovic 4-6 6-3 2-0 when the Serbian retired.
Canadian wildcard Bianca Andreescu, 17, claimed a surprise win, overcoming Camila Giorgi 5-7 6-3 6-4.
Andrea Petkovic, the 2013 runner-up, edged to a 6-4 7-6 (7-5) victory against Kurumi Nara.
Sara Errani beat Risa Ozaki 7-6 (7-0) 6-2 and Mariana Duque-Marino eased past Shelby Rogers 6-4 6-1.
Alison Van Uytvanck also booked her spot in the second round thanks to a 7-5 0-6 6-1 win over Jana Cepelova.