It was the Asian nation's first World Cup victory over New Zealand following 12 defeats. Replying to New Zealand's 6-150 they made 5-153.
Nilakshika Silva (54no off 37 balls) and Kaushini Nuthyangana (24no off 14) combined for an unbeaten 48 off 28 balls to reach the target with two balls remaining at the Rose Bowl.
New Zealand were again undone by dropped catches; six in the opening loss to the West Indies being followed by three more against Sri Lanka, two of them key.
Sri Lanka captain Chamari Athapaththu was put down on six and made 27 off 19 to start their charge in the powerplay, and Silva was dropped on one by Bree Illing, a simple chance at short fine leg off Melie Kerr.
"New Zealand is one of the best teams and the world champions, and the tournament favourites," Athapaththu said. "We beat them and it means we can beat anyone in the comp, we just need to execute our plans."
An unchanged New Zealand chose to bat first but only captain Kerr and Sophie Devine got going with 45s.
The two were battering Sri Lanka together until Kerr was caught on the boundary. Devine gave New Zealand hope but she was caught on the boundary, too, in the 18th over, and the Kiwis collected only 11 runs off the last two overs.
A run of four Sri Lanka wickets in four overs for ten runs, to be 4-55, looked to have ended their chase but Silva was dropped and turned her life into a player-of-the-match performance.
When Kavisha Dilhara ran herself out at 5-105 in the 15th, Sri Lanka appeared to be on the ropes. But Silva's second T20 fifty in 13 years and Nuthyangana's career-best rushed Sri Lanka to the win after an opening hiding from England.
