No.1 seed Jabeur eased past American Alycia Parks, the world No.313, 6-4 6-0 in Charleston on Friday (AEST).
Eyeing her first WTA Tour title, Tunisian Jabeur will face Nao Hibino for a place in the semi-finals.
Colombia's Osorio Serrano, who claimed her maiden singles title at the Copa Colsanitas in Bogota last week, rallied past Christina McHale 2-6 7-6 (7-2) 7-6 (7-1).
Osorio Serrano and Clara Tauson will go head-to-head in the quarters at the WTA 250 event.
Linda Fruhvirtova, the 15-year-old Czech teen, and Astra Sharma also advanced at the expense of Emma Navarro and No.8 seed Madison Brengle.