Dodin, seeded second in the tournament, was one game from defeat but fought back to squeeze out a 5-7, 7-5, 6-4 victory over Canada's Gabriela Dabrowski.
The Frenchwoman trailed 5-4 in the second set with Dabrowski serving for the match, but she scored a crucial break to win the final three games and force a decisive third set.
Dodin advances to the second round to face American Caroline Dolehide, who had little difficulty in beating Canadian Charlotte Robillard-Millette, 6-3, 6-2.
Third-seeded Timea Babos endured two difficult sets before finishing off Jamie Loeb 7-6 (7-4), 5-7, 6-2 in her first-round victory.
After losing to Maria Sharapova in the second round of the U.S. Open, Babos struggled early and even trailed by a break point in the second set. In the third set, the Hungarian rolled past her American opponent.
Babos improved her hardcourt record to 12-13 and after beating Loeb, she is 8-4 against opponents ranked 101 or lower.
Babos will face Britain's Naomi Broady in the second round. Broady also needed three sets to get a 6-4, 3-6, 6-3 victory over Australian teenager Destanee Aiava.
Broady secured her win by recording 13 aces and losing serve only once.
New Zealand's Marina Erakovic also advanced to the second round with a 6-4, 6-1 victory over Russian qualifier Alla Kudryavtseva. Erkaovic will face seventh-seed Alison Van Uytvanck or wild card Carol Zhao.