Chasing a target of 165, Sri Lanka fell prey to Kumar who first took out opener Avishka Fernando before dismissing Chamika Karunaratne, Isuru Udana and Dushmantha Chameera in the final four overs to guide India to a winning start in the series.
"It was a good batting wicket but there was some help for bowlers too," said Kumar, who finished with 4-22.
India's Deepak Chahar (2-24) also troubled Sri Lanka, who were bowled out for 126 runs with nine balls to spare.
After being sent in to bat, India got off to a shaky start with debutant Prithvi Shaw dismissed on the first ball and Sanju Samson out for 27. But a half-century from Suryakumar Yadav and skipper Shikhar Dhawan's 46 helped them set a decent target.
Sri Lanka Cricket said the press box for Sunday's match was closed after one of their members working in that area tested positive for COVID-19.
India have named a depleted squad for their tour of Sri Lanka because several of their frontline players are in England preparing for a five-test series against Joe Root's side starting on Aug 4 in Nottingham.
The second T20 international will be held on Tuesday followed by the third on Thursday, with both games also being played in Colombo.