"Maybe previous years I couldn't win the matches in the third set, so I'm really happy that I changed this to myself, and I feel stronger," Halep told reporters.
"So when I go to the third set, I don't panic. I'm relaxed and I'm just fighting till the end, and I'm sure that I have a big chance to win the match, so I just go for it."
An upset looked possible in the first set when Dodin's big first serve troubled Halep, who was crushed by eventual champion Naomi Osaka in the Indian Wells semi-finals last week.
But the defensive-minded Halep, who at times reached down to rub a troublesome right ankle, improved as the match wore on, getting the better of France's Dodin in extended rallies and in the key moments.
The match was in the balance until Halep had break points at 5-5 in the third set and jumped all over a weak second serve from the 21-year-old Dodin to move ahead.
Halep won the next game to love, rocketing down an ace on match point to set up a meeting with 30th-seeded Pole Agnieszka Radwanska, who defeated Belgian Alison Van Uytvanck 6-3 7-6(4).
"I felt just that I played bad, but it's a good day at the office and I'll take it," Halep said.
"You have these days, and when you don't give up, it's the most important thing. I think that's why I won today, because I didn't give up."
In an evening match, American 14th seed Madison Keys retired with what appeared to be a left hamstring injury early in the second set against three-times champion Victoria Azarenka, who led 7-6(5) 2-0.
Keys had the leg taped by a trainer after losing a first set that stretched to over an hour, but lasted just two more games before calling it quits.
Among the seeds exiting the tournament were number seven Carolina Garcia of France and number 12 Julia Goerges of Germany.
(Reporting by Rory Carroll in Los Angeles and Andrew Both in Cary, North Carolina; Editing by Ed Osmond/Peter Rutherford)
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