Despite dropping her serve in the opening game of the second, Sevastova, seeded 16th, rallied back to force a decider.
Sharapova, who had never lost a third-set tiebreak, needed a brave forehand winner at 6-5 before sealing a second-round spot in just over three hours.
She will take on the winner between American qualifier Jennifer Brady and Ekaterina Makarova of Russia in the second round.
Earlier in the day, second seed Simona Halep snapped her three-match losing streak with a hardfought 6-3 3-6 6-2 win over American Alison Riske.
The Romanian, who had not won a match since her defeat in the Cincinnati final to Garbine Muguruza, overcame a sloppy start before claiming 12 of the final 14 points to set up a second round clash against Magdalena Rybarikova or Eugenie Bouchard.
Germany's Angelique Kerber began her campaign with a 6-2 7-5 win over Japanese teenager Naomi Osaka.
Kerber, the former world number one, came back from a break down to win five consecutive games in the second set to seal a second-round spot.
Twice Wimbledon champion Petra Kvitova, seeded 12th, produced an impressive array of powerful groundstrokes to beat fellow Czech left-hander Kristyna Pliskova 6-3 7-5 in the opening round.
(Reporting by Hardik Vyas in Bengaluru, editing by Ed Osmond)
Share
