Romanian Buzarnescu, who is still seeking her first win of the 2019 season, was undone by 13 double faults, and allowed 2016 Doha runner-up Ostapenko to have 20 break-point opportunities, of which the Latvian converted seven.
After dropping the second set, Ostapenko regained the momentum by saving four break points in the opening game of the decider.
Buzarnescu conceded a 3-1 lead when she committed her 12th double fault of the day on break point, before Ostapenko unleashed a string of forehand winners to seal the victory in an hour and 45 minutes.
Ostapenko, who won the French Open in 2017, will next face world number seven Elina Svitolina.
"I hope the wind is not going to be the same tomorrow as it was today because it was very tough to play," Ostapenko said.
"Otherwise, I mean, (Svitolina's) a very consistent player. I will have to play on a very high level to beat her, of course."
Fifth seed Kiki Bertens recovered from a nightmare opening set to beat Italian Camila Giorgi 0-6 7-6(7) 6-4.
Giorgi produced an impressive display of power-hitting and finished with 35 winners but fell short as her Dutch opponent won the critical points in the final two sets.
Bertens won 80 percent of her first-serve points in the final set to clinch victory and set up a second round clash with 2016 Doha champion Carla Suarez Navarro.
Latvia's seventh seed Anastasija Sevastova suffered a shock opening round exit as she was beaten 7-6(5) 6-4 by Russian qualifier Anna Blinkova.
It was a career-best performance for the 20-year-old Blinkova, whose previous two top-30 wins also came in Doha last year.
Earlier in the day, former world number one Caroline Wozniacki withdrew from the tournament due to a viral illness.
Australian Sam Stosur, who replaced Wozniacki in the main draw as a lucky loser, fell at the first hurdle after a 6-4 6-2 defeat by Czech qualifier Karolina Muchova.
(Reporting by Hardik Vyas in Bengaluru; Editing by Toby Davis)
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