Hingis, looking to reclaim her title at the season-ending tournament, stormed into the round of eight as she dumped Julie Halard-Decugis of France, 6-2, 6-3, in the first evening match.
Kournikova spoiled Jennifer Capriati's return to this event in the late match as the seventh seed from Russia recorded a 6-4, 6-4 victory, her first at this tournament.
Earlier today, South Africa's Amanda Coetzer easily won her first match at this event since 1993 as she upended No. 8 Chanda Rubin of the United States, 6-2, 6-1, to advance to the quarters.
In a match pitting the top-ranked singles player against the top-ranked doubles player, Hingis displayed an almost flawless performance in improving her season record to 74-10. She made only three unforced errors during the 51-minute match, breaking her French opponent four times and landing 85 percent of her first serves.
"I knew that I had to serve well if I want to do well, especially if I want to do well on this surface (carpet)," Hingis said. "It just puts me in better position if you start putting in the first serve. I know that if I win this tournament, that is going to be a big part of my game, serving well."
Hingis did not win a Grand Slam title this season for the first time since 1996, making her lone appearance in a Grand Slam final at the Australian Open in January. Although she has not won a major this year, the 20-year-old from Switzerland has won a tour-high eight titles.
Hingis has played in the final at the WTA Tour's season-ending tournament three times, including the last two years. She won in 1998 but lost to Lindsay Davenport last season.
"I'm happy with the way I've played and the way I've constantly improved and raised my level, especially in the second half of the year," Hingis said. "Maybe I didn't win a Grand Slam, but there are many more years to come."
Up next for Hingis will be the winner of Wednesday's match between sixth seed Nathalie Tauziat of France and American Amy Frazier.
The 30-year-old Halard-Decugis committed 18 unforced errors and remained winless in five lifetime meetings with Hingis. She was competing in her second straight Chase Championships in singles and fourth of her career.
After the match, Halard-Decugis announced her retirement after playing 14 years on the tour and winning 12 singles titles.
"It's just a feeling that I want to stop, have a family life and do something else," she said. "I have some very happy years playing tennis. I know that I would be able to play some more years, because physically, I feel very well, but I trust my mind. I want to do something else. It took me a long time to decide. And I'm happy that I have a good choice."
The Kournikova-Capriati match saw both players struggle early with their serve as there were four consecutive breaks from games three through six of the opening set. Kournikova got the crucial break in the ninth game and closed out the set in the next game with an overhead smash.
Both players held serve in the second set until the sixth game, when Capriati broke for a 4-2 lead. But that advantage quickly disappeared in the next game as Capriati fell behind 0-40 after a double fault and got broken three points later.
Kournikova saved a break point and held for 4-4, but Capriati, who committed nine double faults overall, made two straight to start the ninth game and eventually lost her serve when a blistering forehand in the corner by Kournikova forced Capriati to hit a backhand long.
At 5-4, Kournikova served out the match at love to prevail in just over an hour for her first win here in three tries.
"I'm just really happy with the way I played, and I definitely was going into the match with a lot of confidence," said Kournikova, who improved to 3-1 lifetime against Capriati. "I've beaten her before and I know exactly how to play her. I knew that she is not in her best shape and that I am playing well, so I was pretty confident before the match."
Still searching for that elusive first singles title, Kournikova will take on fourth seed Conchita Martinez of Spain in the quarterfinals. She is seeded second with Hingis in doubles.
Capriati was making her first appearance at the Chase Championships since 1992 after battling off-the-court problems in the mid 1990s. She continued her comeback this year by reaching her first Grand Slam semifinal since 1991 at the Australian Open, and claimed her only title of the year at Luxembourg last month.
"It was good to be in there again," said Capriati, a two-time quarterfinalist here. "But it is a little disappointing because I wanted to play better. I felt good coming into the match, but some things just didn't go my way."
Coetzer had dropped six consecutive matches here since advancing to the quarterfinals in 1993. She claimed her only title of the year at Antwerp in May and reached the final at the German Open earlier that month.
Rubin was making her first appearance at the season-ending event since 1995. She entered this event with some momentum, having won her first title in almost two years two weeks ago in Quebec City.
This is the 30th edition of the world championship for women's tennis and the 24th and final time the tournament will be held at Madison Square Garden. The WTA has chosen to move the event to Munich, Germany starting next year, citing financial reasons.