The paceman also sent back Shikhar Dhawan as India, top-ranked in the shortest format, continued losing wickets trying to play high-risk shots.
Shanaka, playing only his second match for Sri Lanka, sent back Suresh Raina and captain Mahendra Singh Dhoni in one over to dent India's chances of posting a competitive total.
Ravichandran Ashwin top-scored with an unbeaten 31 to take India past the three-figure mark, which looked a distant prospect at one stage with the hosts struggling at 72 for eight.
Ashish Nehra picked up two early wickets to give India a glimmer of hope but a 39-run stand between captain Dinesh Chandimal (35) and Chamara Kapugedera (25) put the tourists back on course.
Sri Lanka, missing fast bowler and regular captain Lasith Malinga, all-rounder Angelo Mathews and spinner Rangana Herath in the series due to injuries, stuttered once again with off-spinner Ashwin picking up two wickets.
But Milinda Siriwardana hit 21 off 14 balls as Sri Lanka reached their target with two overs to spare.
The second match of the series will be played in the eastern Indian city of Ranchi on Friday.
(Reporting by Sudipto Ganguly in Mumbai; editing by Justin Palmer)
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