Capriati, the second seed, required three sets to get past Russian teenager Nadia Petrova, 6-3, 3-6, 6-2, in their first career encounter.
Davenport, the third seed, had a much easier time beating South African Amanda Coetzer, for the 10th time in their 13 lifetime meetings, 6-3, 6-4.
Capriati, who on Monday became just the ninth player in the history of the computer rankings to ascend to the top spot, continues to enjoy a career-best season. The American claimed the Australian and French Open titles and reached the semifinals at Wimbledon and the U.S. Open.
The 25-year-old also won the Family Circle Cup at Charleston, S.C. in April and advanced to the finals at Oklahoma City, Miami, Berlin and Toronto.
Capriati awaits the winner of Thursday's match between a pair of qualifiers, Switzerland's Marie-Giane Mikaelian and Russia's Tatiana Panova.
Davenport, who last sat atop the WTA rankings on May 21, 2000, has won 12 of her last 13 matches at the Zurich event. She captured consecutive titles in 1997 and 1998 before losing a year ago in the final to Martina Hingis of Switzerland, who withdrew from this week's event due to an ankle injury.
The Californian won her fifth tournament of the year and the 35th of her career at last week's Porsche Tennis Grand Prix at Filderstadt, Germany.
Davenport has posted a 31-5 record since returning in June after a three-month layoff due to an injured right knee. The 25-year-old American has advanced to the final in five of those eight events, winning three titles.
Davenport will face either Slovak qualifier Daniela Hantuchova or Austrian Barbara Schett in the next round.
In other action, fifth seed Nathalie Tauziat of France cruised past Iroda Tulyaganova of Uzbekistan, 6-2, 6-4, and eighth seed Silvia Farina Elia of Italy coasted to a 6-2, 6-4 victory over Lisa Raymond of the United States.
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