Seeded 26th, Sania was made to sweat by her unheralded opponent for about one hour and 40 minutes before the Indian came through.
Sania, the World number 27, now meets 60th ranked Laura Granville in the second round, after the American routed Frenchwoman Olivia Sanchez 6-2, 6-2.
The Hyderabadi began well by pocketing the first set, conceding just two games, before the Estonian hit back. Sania did manage to stretch the second set to a tie-breaker but, Kanepi had the last laugh as stayed afloat and forced a decider.
In the final set, Sania was in her elements again and broke the Estonian twice. She dropped just one game in the set and was serving for match when Kanepi buried a return into the net and bowed out.
World number one Roger Federer breezed over qualifiers into the second round, while reigning Grand Slam champions Serena and Venus Williams also advanced, adds Reuters.
Cypriot Marcos Baghdatis, the 18th seed, was the top-ranked player to lose on day one, losing to Max Mirnyi of Belarus 6-3, 7-5, 3-6, 7-6 (8/6).
Three-time defending champion Federer, seeking his 12th Grand Slam singles crown, opened the year’s final Grand Slam event on Monday with a 6-3, 6-2, 6-4 victory over 320th-ranked American Scoville Jenkins. Eighth seed Serena Williams, who won her eighth Grand Slam title in January at the Australian Open, showed some rust from a long layoff due to a thumb injury but defeated Germany’s Angelique Kerber 6-3, 7-5.
Reigning Wimbledon champion Venus Williams ripped Hungarian qualifier Kira Nagy 6-2, 6-1 in 54 minutes. The 12th-seeded American, a six-time Grand Slam winner, fired a main-draw women’s record 129-mph serve in the first set.
Venus Williams next plays Romania’s Ioana Raluca Olaru, who ousted American Ashley Harkleroad 7-5, 1-6, 6-4. Serena Williams will next meet Italy’s Maria Elena Camerin, who beat Russian Tatiana Poutchek 6-3, 7-6 (7/4). Federer, 26, seeks his 12th Grand Slam title in pursuit of the all-time record 14 won by Pete Sampras and his fourth US Open crown in a row. Not since Bill Tilden’s run from 1920 to 1925 has anyone won four consecutive US Opens.
The second round foe for Federer is another qualifier, Paul Capdeville. The Chilean outlasted Germany’s Rainer Schuettler 5-7, 6-3, 6-7 (4/7), 6-4, 6-3.
Hingis wins
Sixteenth seed Swiss player Martina Hingis entered the second round with a 6-0, 6-3 win over Frenchwoman Mathilde Johansson.
Serbian third seed Jelena Jankovic advanced in straight sets, as did Serbian fifth seed and French Open runner-up Ana Ivanovic.
Jankovic, 22, was nagged by a damaged left shoe in the second set but won her women’s best 68th match of the year. She subdued Slovakian Jarmila Gajdosova 6-2, 7-6 (7/2) after 80 minutes.
Former world number one Lleyton Hewitt made an impressive start to his campaign as he outclassed American Amer Delic 6-2, 6-4, 6-2 to reach the second round.
Former champion Svetlana Kuznetsova battered her way into the second round with a 6-2, 6-3 victory over Czech Klara Zakopalova.