Hingis, the defending champion and four-time winner here, was blanked 0-6 in the opening set. She pulled herself together and took the next two sets 6-4, 6-2 to improve to 5-0 all-time against the 26-year-old Bulgarian.
Hingis continues to attempt to end a winless streak of 12 tournaments, the longest drought of her career. She may lose her No. 1 ranking to Jennifer Capriati if she does not perform well this week.
Maleeva also was eliminated by Hingis in the 1997 quarterfinals here, which was her best showing at this tournament.
Capriati, the second-seeded American star, had an easier time in her second-round match against Dutch qualifier Miriam Oremans, 6-1, 6-4.
It was the second time Capriati defeated Oremans in straight sets this year. She won 6-0, 6-2 at the Australian Open en route to the title. Capriati is looking to capture her 13th career title.
Also in the second round, No. 5 Amelie Mauresmo of France went the distance against South African Patty Schnyder, 3-6, 6-3, 7-5 in a second-round match.
Kournikova committed 25 double faults and dropped to 0-3 since returning to action at Leipzig, Germany last month. Kournikova did not compete on the tour between February and July due to a stress fracture on her left foot.
Sandrine Testud of France completed the first round with a 6-2, 6-2 victory over Slovenian qualifier Maja Matevzic. Testud faces Magui Serna of Spain in the second round.