Third seed Mara Santangelo also advanced, along with China's Peng Shuai and Austrian Sybille Bammer.
Sania, the fifth seed, struggled to produce her best tennis in windy and humid conditions, and twice double-faulted on set point when leading 5-3 in the first set.
After a total of four double-faults she conceded the game with a wild forehand error, but then broke for the set with a winning return off a second serve.
Tanasugarn continued to press in the second set by moving the ball around well and hitting some strong ground strokes to build a 5-1 lead.
But Sania once again gained the upper hand in the deciding set, breaking to lead 3-1 by forcing a backhand error and then giving up just five points in the final three games against a tiring opponent.
Tanasugarn criticised the scheduling after she completed her first-round match at 12.30 am, claiming that if she had played on Thursday the outcome would have been different. But that was of no concern to Sania.
“I obviously didn't play close to my best, but under the conditions and circumstances it was a good match to win,” said Mirza.
Italy's Santangelo dismissed Alla Kudryavtseva of Russia 6-2, 6-4 to book her place in the last eight. Santangelo quickly exploited her opponent's weak serve and allowed the Russian just four points in the opening three games.
Kudryavtseva held off a break point at 0-3, and then had an opportunity to come back when Santangelo double-faulted at 4-2 to give her break point. But the Italian held off the threat with a forehand volley, held for 5-2, and broke again for the set.Kudryavtseva failed to take advantage of another opportunity early in the second set when she held two break points to lead 2-1, and she then dropped her own serve in the next game to fall behind 3-1.
Yet another chance went begging after Santangelo served for the match at 5-3 but dropped her serve with a double-fault and, although the Russian came within two points of levelling at 5-5, Santangelo broke once more for the match.
China’s Peng dropped the first set against Italy’s Romina Oprandi, before dominating the rest of the match to win 6-7 (5/7), 6-1, 6-3.
Meanwhile, the left-handed Bammer progressed with a 6-2, 7-5 victory over Anastassia Rodionova of Russia.
Results (x denotes seeding):
Second round
Mara Santangelo (ITA x3) bt Alla Kudryavtseva (RUS) 6-2, 6-4; Sania Mirza (IND x5) bt Tamarine Tanasugarn (THA) 6-4, 2-6, 6-1; Sybille Bammer (AUT) bt Anastassia Rodionova (RUS) 6-2, 7-5; Peng Shuai (CHN x4) bt Romina Oprandi (ITA) 6-7(5) 6-1, 6-3. –AFP