Dinusha made a stellar 133 not out, his second hundred of the match and third in four innings, to power Sri Lanka to 429-9 while following on when the players shook hands.
India won the series 1-0 but Sri Lanka will take a lot of heart from their fightback at the Sinhalese Sports Club Ground.
Dinusha forged a century-run stand with Keshara Nuwantha to frustrate India and stayed put to help snatch a memorable draw.
"When the team is down, I wanted to get runs for the team," player of the match Dinusha said. "Getting runs against India is a pleasure as a player."
Earlier, Sri Lanka resumed the final day of the rain-plagued match just 16 ahead and with only four wickets in hand.
They needed to bat out of their skin to save the test match, and Dinusha and Nuwantha (46) did exactly that, denying India success throughout the morning session.
Dinusha's maiden test hundred had been the bedrock of Sri Lanka's first-innings total of 290, and the 25-year-old followed it up with his fourth successive score of 50 or more to keep his side alive in the contest.
Frustration grew in the Indian camp as Dinusha and Nuwantha put together a defiant 112-run stand, leading a spirited resistance.
Reviews favour Nuwantha
Nuwantha also found himself on the right side of the review system twice, including in the final over before lunch, when he successfully overturned an lbw decision.
Spinner Manav Suthar finally broke the partnership in the first over after the interval. His delivery bounced and kissed the shoulder of Nuwantha's bat before carrying through to Dhruv Jurel at slip.
Jurel then took a second catch to dismiss Prabath Jayasuriya for two, but Dinusha remained unperturbed despite being left to bat with the tail.
Lahiru Kumara provided invaluable support down the order, absorbing 90 balls for his dour 12.
On a pitch offering little assistance to the seamers, Dinusha defended with assurance and swept with panache to continue defying India.
Touring captain Shubman Gill rotated his bowlers and continually adjusted his field settings but nothing appeared capable of unsettling Dinusha, who batted with tremendous discipline.
Gill stressed that India tried all they could in the match.
"We fielded for about 250 overs, for three days consecutively, and we did everything we could, but they played really well," said the India captain.
"Their batters showed a lot of skill and they had a little bit of luck going their way.
"A lot of positives for us, including learning from the opposition. They were under a lot of pressure and the way they came back was tremendous to see."
India top-order batter Devdutt Padikkal was adjudged player of the series for his hundreds in Galle and Colombo.