Sri Lanka, who last week lost in the semi-finals of the Champions Trophy to eventual winners India, had looked on course for a healthy total, reaching 104 for two in the 22nd over after being put into bat by the host nation.
However, spinner Narine (four for 40) and paceman Rampaul (three for 38) combined to rip out the Sri Lanka middle order and the tourists were dismissed with nine balls to spare.
Sri Lanka captain Angelo Matthews had sought to build on the good early work of Mahela Jayawardene, who struck a run-a-ball 52 at the top of the innings, but he could not find sufficient support from his team-mates as he was left stranded 55 not out.
Dwayne Bravo, having also claimed the first wicket of the day when he had Upul Tharanga caught behind, wrapped up the Sri Lanka innings with last man Ajantha Mendis snapped up at first slip to finish with figures of two for 37.
India are the other team involved the tournament.