Spinners Rinku Singh and Suryakumar bowled the 19th and 20th overs in Sri Lanka's chase of 137 and the match ended in a tie at Kandy's Pallekele International Stadium.
Washington Sundar then took two Sri Lankan wickets to end the Super Over quickly with the hosts scoring just two runs, which India overhauled in just one hit by Suryakumar -- in batting mode.
Sundar was elected man of the match after he picked up two wickets in the Sri Lankan chase. Ravi Bishnoi, Rinku and Suryakumar also took two each.
India's new-age T20 team, led by Suryakumar and coached by Gautam Gambhir, have started strongly after the retirements of stalwarts Rohit Sharma, Virat Kohli and Ravindra Jadeja following the team's T20 World Cup triumph last month.
Suryakumar led from the front with the bat in the first two wins to clinch the series.
He surprisingly bowled the final over when Sri Lanka needed six to win.
"More than the last over, I feel when we were around 30-4, how the boys showed character, 140 was a par score on that track," player-of-the-series Suryakumar said. "I told, if we put our heart in, we could pull it off."
He added: "The amount of skill they have, self-confidence, it makes my job easy. The positivity, the care for each other is unbelievable."
Suryakumar, an attacking batsman who bowls off-spin, took two wickets in successive balls in his first outing as a bowler in T20 matches after Rinku also got the ball in the penultimate over for the first time in the shorter format.
- Batting collapse -
It was another batting collapse for Sri Lanka who fell like a pack of cards after they needed 30 off the last 30 balls with nine wickets in hand.
Kusal Perera made 46 before being caught and bowled by Rinku.
Sri Lanka, led by Charith Asalanka, won the toss delayed due to rain in Kandy and elected to field first on a new pitch, which helped spinners.
Maheesh Theekshana took charge in the second over to send back Yashasvi Jaiswal for 10 and soon India were reduced to 30-4 inside six overs with Suryakumar out for eight.
Debutant Chamindu Wickramasinghe took his maiden wicket when he dismissed Sanju Samson for a duck.
Shubman Gill, who made 39, and Riyan Parag, who hit 26, attempted to rebuild in a sixth-wicket partnership of 54.
Sri Lanka kept chipping away but Sundar boosted the total with his 25 and then the visiting bowlers roared.
"Definitely (the target was gettable), we need to do more as professional cricketers," said Asalanka. "I want to see good batting performances in the ODIs. Our top three and four have done well and that's the only positive for us."
India had rested top players, including Hardik Pandya and Rishabh Pant, in four changes to the team.
The two teams will now play three ODI matches as India await the return of skipper Rohit and Kohli in the 50-over series starting August 2.