Gauff, 15, beat world No. 8 Bertens in straight sets 7-6(1), 6-4 in a closely fought match to advance to the final four.
The American matched Bertens in the first set before dominating her opponent in a tiebreak. Gauff broke Bertens' serve early in the second set and kept her composure throughout the remainder of the set to emerge victorious.
Gauff, who made an odds-defying run to the Wimbledon last 16 in July, will provisionally climb into the top 100 when the WTA rankings are released next week, edging closer to automatic qualification for the Australian Open in January.
Gauff beat Stefanie Voegele of Germany in the round of 32 after she stepped in as a lucky loser for injured sixth seed Maria Sakkari of Greece. The American had initially failed to make the main draw of the tournament, losing in straight sets to Germany's Tamara Korpatsch in the second round of qualifying on Monday, but is now playing in her first tournament since advancing to the third round of the US Open, where she fell to then world No. 1 Naomi Osaka.
In the other side of the draw, Russia's Ekaterina Alexandrova defeated Kristina Mladenovic 6-2, 6-4 to book her spot in the final four, while Jelena Ostapenko of Latvia beat Kazakhstan's Elena Rybakina 7-5, 6-1.