The world number eight dropped her opening service game as Van Uytvanck took the first set 6-4.
But Bertens battled back to win 4-6 6-3 6-3 and book a last-eight showdown with 15-year-old Coco Cauff.
Highlights: Bertens digs deep to set up meeting with Gauff
Errani accuses wheelchair-bound Bertens after extraordinary match
Earlier in the day, fourth seed Donna Vekic was sent packing after an impressive comeback from Kristina Mladenovic, who won 3-6 6-1 6-2.
Both players struggled severely on serve in the opening set. Mladenovic did not hold once, while Vekic recovered from dropping her first three service games to take the first set 6-3.
Remarkably, Mladenovic turned her fortunes around, holding serve every time going forward while breaking Vekic four times to take the next two sets and advance to the next round.
Viktoria Kuzmova also enjoyed a comeback victory, battling to a 2-6 6-2 6-4 win over ninth seed Anastasia Pavlyuchenkova.
Eighth seed Ekaterina Alexandrova did manage to halt Laura Siegemund’s riposte, winning 7-6(1) 2-6 6-4 after more than two-and-a-half hours on court.
Jelena Ostapenko reached the quarter-finals after opponent Alize Cornet retired trailing 7-5 4-1.
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Heartbreaking moment as tearful Zavatska runs out of rackets mid-match