Burel, 20, a former world number one in the junior rankings who is coached by French former tennis star Thierry Champion, was playing in her first final on the WTA tour.
The 25-year-old is the first Czech to lift the Suzanne Lenglen Cup since Hana Mandlikova, who represented Czechoslovakia, triumphed in 1981.
Krejcikova won six games in a row to take the opening set after dropping serve in the first, before Pavlyuchenkova gained the upper hand in the second set.
Zidansek advanced 7-5, 4-6, 8-6 and will play Kazakh 21st seed Elena Rybakina or 31st-seeded Anastasia Pavlyuchenkova of Russia for a place in Saturday's final.
"We didn't expect for this to happen, Bartoli told ITV Sport during her role as a pundit on Friday. "We didn't think the situation was so bad with Naomi. I think the WTA is trying really hard to find some solution.
Gauff eased through the opening set after jumping out to a 3-0 lead but did not have things her own way after that against the player she beat to win the Parma title last month.
Kenin made 18 unforced errors in an unconvincing first-set display but broke decisively in the 11th game with her sixth consecutive point as she bagged seven games in a row to open up a 4-0 lead in the second.