Seeded 26th, Sania was made to sweat for one hour and 22 minutes and the win pits the Indian against sixth seeded Anna Chakvetadze in the third round after the Russian beat Australian Nicole Pratt 6-3, 6-4 in another second round match.
Sania and Laura exchanged two breaks in the first set before the Indian broke loose and converted her third break point to go one up in the match.
Sania began well in the second set and looked in a hurry to wrap up the match as she raced to a 4-2 lead. Just then the American threw in the spanner, winning two of the next three games to reduce the gap.
Sania, however, still had the advantage as she went into the 10th game, serving for the match. She was leading 30-0 before squandering two match points, which allowed the American to break her and level it 5-5.
That served a wake up call for the Indian, who raised her game and bagged the next two games to seal the issue.
Meanwhile, former champion Svetlana Kuznetsova overcame a second-set lapse to beat Camille Pin of France 6-3, 4-6, 6-0 to reach the third round .
The 2004 champion, seeded fourth, led 3-1 in the second set but lost five of the next six games before recovering to storm through the decider.
In the other matches, Belgian top seed Justine Henin needed just over an hour to defeat Bulgarian qualifier Tsvetana Pironkova 6-4, 6-0.
Former champions Venus and Serena Williams both started slowly before hitting their stride and winning their second-round matches in straight sets.
Venus eased past Romanian Ioana Raluca Olaru 6-4, 6-2 while Serena defeated Italian Maria Elena Camerin 7-5, 6-2.
Gasquet pulls out
In the men’s section, French 13th seed Richard Gasquet pulled out, because of illness.
The Wimbledon semi-finalist said he was suffering from a fever and a sore throat. His withdrawal handed American wildcard Donald Young a walkover into the third round. Earlier, Rafael Nadal advanced to the third round , but a knee injury could hamper his chances of dethroning top-ranked Roger Federer.
The Spaniard battled through the pain and an inspired performance by Australian Alun Jones before winning 7-5, 3-6, 6-4, 6-1 in a two-hour, 45-minute affair in Arthur Ashe Stadium. Second-seeded Nadal hurt his left knee during practice three days ago and had the trainer wrap the joint after his service was broken by Jones in the third set for a 4-3 deficit.
Meanwhile, Briton Andy Murray continued his recovery from a wrist injury by scrapping his way to a 5-7, 6-3, 6-1, 4-6, 6-1 victory over veteran Swede Jonas Bjorkman.
The 19th-seeded Scot, who returned earlier this month after 12 weeks out, needed three hours 37 minutes and 16 aces to clinch a meeting with 14th-seeded Argentine Guillermo Canas or Hyung-Taik Lee of South Korea.
Results:
Men's singles (prefix denotes seeding): Second round: 19-Andy Murray (Bri) bt Jonas Bjorkman (Swe) 5-7, 6-3, 6-1, 4-6, 6-1; 9-Tomas Berdych (Cze) bt Simone Bolelli (Ita) 7-5, 6-3, 6-3; Fernando Verdasco (Spa) bt Gilles Simon (France) 6-0, 6-4, 6-3; Donald Young (US) w/o13-Richard Gasquet (Fra);  1-Roger Federer (Sui) bt Paul Capdeville (Chi) 6-1, 6-4, 6-4; John Isner (US) bt Rik De Voest (SA) 6-3, 6-3, 7-6(4). First round: Tim Henman (Bri) bt 27-Dmitry Tursunov (Rus) 6-4, 3-6, 6-3, 6-4; Teimuraz Gabashvili (Rus) bt 7-Fernando Gonzalez (Chi) 6-4, 6-1, 3-6, 5-7, 6-4; 2-Rafael Nadal (Spa) bt Alun Jones (Aus) 7-5, 3-6, 6-4, 6-1; Mardy Fish (US) bt Bruno Echagaray (Mex) 6-4, 6-1, 6-2; 17-Carlos Moya (Spa) bt Bjoern Phau (Ger) 6-1, 6-2, 6-4; 8-Tommy Robredo (Spa) bt Bobby Reynolds (US) 6-3, 7-6(5), 6-1.
Women (second round): 16-Martina Hingis (Swi) bt Pauline Parmentier (Fra) 6-2 7-5; 11-Patty Schnyder (Swi) bt Severine Bremond 6-3, 6-0;  (France) 6-3, 6-0; Julia Vakulenko (Ukr) bt Jelena Kostanic Tosic (Cro) 6-2, 6-1; 4-Svetlana Kuznetsova (Rus) bt Camille Pin (France) 6-3, 4-6, 6-0; Tamira Paszek (Aus) bt 24-Francesca Schiavone (Italy) 6-3, 7-5; 31-Anabel Medina Garriques (Spa) bt Lourdes Dominguez Lino (Spa 7-5, 6-2; 26-Sania Mirza (Ind) bt Laura Granville (US) 6-3, 7-5 ; 6-Anna Chakvetadze (Rus) bt Nicole Pratt (Australia) 6-3, 6-4; 18-Shahar Peer (Isr) bt Bethanie Mattek (US) 6-2, 6-1; 8 -Serena Williams (US) bt Maria Elena Camerin (Ita) 7-5, 6-2; 27-Vera Zvonareva (Rus) bt Elena Likhovtseva (Rus) 6-4, 6-2; Ahsha Rolle (US) bt Karin Knapp (Ita) 3-6, 6-4, 6-3.