Kournikova, whose spectacular game has led some to believe she could one day become the world No 1, let three match points slip away before beating Hingis, 6-3, 7-6.
"I'm been praying this win would come," Kournikova said. "I'm just very very happy." Hingis, the world's No 1 player, had knocked the 16-year-old Russian out of three of the past four Grand Slam events, losing just one set in their four previous meetings.
Jana Novotna, the third seed, also advanced into the semi-finals with a 6-3, 5-7, 6-3 win against Romania's Irina Spirlea. She will face the 18-year-old French qualifier Amelie Mauresmo, who beat Austria's Barbara Paulus, 6-4, 6-2.
Earlier, Hingis had dismissed the idea that Kournikova was another emerging rival for her No 1 status. "I know she can play, but she's never shown it against me," Hingis said. "She was always too nervous."
Kournikova has risen from the No 32 in the world to No 16 this year, beating four top 10 players at the Lipton Championships to reach her first final. But the Russian's path to the top has often been blocked by Hingis.
But this time Kournikova was prepared against the Swiss player, using her power to decide some spectacular rallies. "I didn't really think about who was on the other side of the net," said Kournikova. "I just played my game. I think she was surprised by that."
Hingis shrugged off the loss just two weeks before the French Open. "It's nothing to cry about. I'll be ready for the French Open," she said.
l Monica Seles' father, Karoly, has died following a long battle against cancer at the age of 64. Of Hungarian origin but from the Yugoslav town of Ujvidek, he worked as a caricaturist before guiding his daughter's career. The family moved to the United States in the late 1980s.Reuse content