But in statistical terms, this was no doubt Gauff’s most impressive victory. Until Friday, she had never beaten a top-40 player. Bertens, a powerful 27-year-old from the Netherlands, is No. 8.

The victory made Gauff the youngest player to reach the semifinals of a WTA tournament since 2004. More important for her longer-term goals, she is guaranteed to break into the top 100 on Monday and has thus secured a spot in the Australian Open, the next Grand Slam tournament. Gauff will be the only player under 18 in the top 100.

“In some corner of her mind, there was some pressure to do really well here to be able to be sure to be in the main draw in Australia,” Jean-Christophe Faurel, one of her coaches, said in an interview from Linz.

Faurel said that helped explain why the week did not start auspiciously: Gauff, who is restricted to a limited number of pro tournaments because of her age, lost in the second round of the qualifying event in Linz.

But she still found a way into the main tournament — as a “lucky loser” when other players withdrew. As she has done repeatedly since qualifying for Wimbledon, Gauff pounced on the opportunity to get the better of her elders.