Making the most of improved batting conditions, the left-hander smashed his way to an unbeaten 174 off 159 deliveries with 19 fours and three sixes, featuring in a 213-run opening stand with the experienced Mahela Jayawardene.
The former captain compiled his 16th ODI century and third against India before becoming the only casualty of the innings, being caught by Umesh Yadav at short third-man off the bowling Ravichandran Ashwin.
It was Jayawardene who set the pace for the innings with another typically fluent effort that occupied 117 deliveries and was embellished by nine fours and two sixes, but by the time he departed in the 39th over, Tharanga was already stepping into overdrive.
He duly reached his 13th ODI hundred - his second against India - shortly after, and with skipper Angelo Mathews (44 not out) promoting himself up the order to turn the screws on their opponents, Indian captain Virat Kohli was at a loss to stem the tide.
With the responsibilities of leadership fully on his shoulders after Mahendra Singh Dhoni was ruled out of the series with a right hamstring injury sustained in their opening match against the West Indies on Sunday, Kohli used seven bowlers, including himself, in a vain effort to thwart the Sri Lankans.
The captain's decision to include Shami Ahmed in the final eleven to open the bowling at the expense of Bhuvneshwar Kumar brought no reward as he conceded 68 runs off his ten overs.
India did not help their cause in the field. Jayawardene was missed at short-third man by Murali Vijay when on 25, while a couple of half-chances also went by and the fielding became increasing ragged as Sri Lanka cashed in to the tune of 180 runs off the last 16 overs of the innings.
Having lost a nail-biter to the hosts two days earlier, the World Cup and Champions Trophy holders face a daunting task when they make their reply as both teams strive for their first points of the competition before the scene shifts to the Queen's Park Oval in Trinidad for the remaining three matches and the final, beginning on Friday when the West Indies entertain India.