Having set Sri Lanka a victory target of 372, India bowled out the hosts for 206 in the post-lunch session on the final day of the contest, minutes before it started pouring at the Galle International Stadium.
Left-arm spinner Suthar (6-55) finished with a 10-wicket match haul as the tourists grabbed a 1-0 lead in the two-match series following their comprehensive victory in India's 600th Test.
Resuming on 4-84, Sri Lanka needed to bat out of their skin to save the match, possibly with some help from the weather.
Skipper Dhananjaya de Silva (59) and Sonal Dinusha (84) struck fighting 50s and defied India with a 95-run stand for the fifth wicket until Ravindra Jadeja separated them.
Once de Silva fluffed his sweep shot against Jadeja, India's victory looked just a matter of time.
Suthar missed a hat-trick but sent down a three-wicket over to send Sri Lanka hurtling towards defeat.
Playing just his second Test, the 24-year-old denied Dinusha his second hundred of the match and clean bowled Prabath Jayasuriya with the next delivery.
Suthar sent back Lahiru Kumara with the final ball of that over and dismissed Keshara Nuwantha in the same fashion to seal India's victory.
"We knew one big shower was going to come and we didn't want to delay the game," India batter Lokesh Rahul said.
"They had one big partnership but we knew if we could break that we could get into their tail.
"It was a great Test match. The wicket played really well all five days."
The second and final Test begins in Colombo on Sunday.
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