The 37-year-old American fell behind 3-1 but recovered with a string of powerful groundstrokes from the baseline, clinching five of the next six games for the opening set.
Peterson, the world number 64, dug deep to grab two break points in the final game of the match but Williams saved both, before wrapping up the win in an hour and 15 minutes.
Williams, a 23-times Grand Slam singles champion, has finished runner-up at two of the last three majors but been unable to add to her tally since returning to the sport following the birth of her first child in 2017.
Williams, who has dropped to number 11 in the world rankings, will next face the winner of the first round clash between her sister Venus and reigning Qatar Open champion Elise Mertens.
After an opening round exit at the Madrid Open last week, former world number one Garbine Muguruza scored her first win in over a month as she eased past China's Zheng Saisai 6-3 6-4.
(Reporting by Hardik Vyas in Bengaluru; Editing by Ian Chadband)
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