Shubman Gill, the India captain, made 80 before retiring hurt, with unbeaten fifties from Washington Sundar and Axar Patel seeing the tourists to a target of 259 with 28 balls remaining as they went 1-0 up in a three-match series.
Victory gave India their first win of a tour featuring T20 series losses to both Ireland and England.
spinner Patel starred with both bat and ball, taking 4-62 and following up with a dominant 57 not out.
Earlier, India reduced England to 80-5 and 107-6 before Joe Root (76 not out) and Liam Dawson (68) shared a seventh-wicket stand of 121.