Asalanka, an off-spinner, bowled the 48th over with India, chasing 231 for victory and needing five from 18 balls at Colombo's R. Premadasa Stadium.
Asalanka trapped a fighting Shivam Dube lbw for 25 as the ninth wicket and then dismissed Arshdeep Singh to bowl out India for 230 in response to the hosts' 230-8.
With no Super Over on offer, the match ended in a tie -- the second such result between the two teams in 50-over format -- and the teams go into the second of the three matches 0-0.
"We felt 230 was enough, should have done better to restrict them further," said Asalanka. "I thought the wicket turned more in the afternoon. Once the light was on it was easier to bat.."
Dunith Wellalage stood out with his unbeaten 67 in Sri Lanka's innings and then took two wickets including skipper Rohit Sharma for 58 with his left-arm spin.
Wanindu Hasaranga, a leg-spinner, and Asalanka took three wickets each to rattle the Indian batting with regular strikes.
KL Rahul, who made 31, and Axar Patel, who scored 33, attempted to put the chase on track in their sixth-wicket partnership of 57 but after they departed India slipped.
Dube, a left-hand batsman, attempted to hit back with his two sixes in his 24-ball knock but failed to take the team over the line.
Big guns return
Rohit and Virat Kohli, who made 24, joined the ODI team after the tourists swept a T20 series 3-0 under Suryakumar Yadav.
"The scores are gettable, you have to bat well to get that," said Asalanka. "We batted well in patches, there was no consistent momentum. We started well but knew the game would start once spin comes on."
Rohit and Kohli retired from T20 internationals after India's World Cup triumph in Barbados in June. They are both part of the ODI and Test set-up with Rohit as leader.
Batting collapses made Sri Lanka suffer in the T20 matches and another scare was on the cards when the home team lost half their side in the 27th over after in-form opener Pathum Nissanka departed for 56.
Wellalage put on key partnerships including a 46-run eighth-wicket stand with Akila Dananjaya, who made 17, to take the attack to the opposition.
Wellalage batted with grit as he picked the gaps and was excellent in running between the wickets to put India's fielders under pressure.
Janith Liyanage and Hasaranga made useful contributions with 20 and 24 runs.
Left-arm quick Arshdeep Singh and left-arm spinner Axar Patel took two wickets each.
Sri Lankan handed an international debut to pace bowler Mohamed Shiraz after quicks Matheesha Pathirana (shoulder) and Dilshan Madushanka (hamstring) were ruled out of the series with injuries.
Shiraz went wicketless in an otherwise inspired bowling by the hosts.
The second match is on Sunday at the same venue.
