The 29-year-old former No. 1-ranked Mauresmo needed 3 hours, 14 minutes to overcome Coin, who also had three match points.
Coin, who got a place in the draw when eighth-seeded Maria Kirilenko of Russia pulled out, ended the 89-minute third set when she pushed a backhand long.
Mauresmo, who won the Australian Open and Wimbledon, and held the No. 1 ranking in 2006, double-faulted on her first match point at 6-4 in the tiebreaker.
"I felt it was long, so long,'' said Mauresmo, who played 3:07 in her win over Mary Pierce at the WTA Championships final in 2005. "It's tough. I'm tight everywhere. A lot of work for my physio.''
She had served for the first set at 5-4 but was broken twice.
In the end, she converted five of her 20 breakpoint chances and fended off 11 of 15 she faced.
"Too many mistakes, too many missed chances - yeah, it was a strange one,'' said Mauresmo, who had to ask how many match points she had but knew about her 10 double-faults.
Mauresmo, now ranked No. 23 and seeded fifth here, will play the winner of Wednesday's night match between top-seeded Ana Ivanovic and Italian qualifier Roberta Vinci.
In an earlier second-round match, Italy's Sara Errani beat Bulgarian Tsvetana Pironkova 6-7 (3), 7-5, 6-3.
On the men's side, Ernests Gulbis lost in the second round 6-3, 6-4 to Paul-Henri Mathieu of France a day after upsetting top-seeded Novak Djokovic.
Florent Serra of France beat Austria's Jurgen Melzer 6-4, 6-3.
Second-seeded Jo-Wilfried Tsonga, who lost last year's Australian Open final to Djokovic, was to play Jarkko Nieminen later Wednesday.
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