The 2007 champions restricted their big-hitting West Indian counterparts to a below-par 129 for seven on a slow track and returned to chase down the target with two balls to spare for their second successive victory.
India lost opener Shikhar Dhawan in the first over to a dubious leg before decision but Kohli (54) and Sharma (62 not out) featured in a 106-run second wicket partnership to take the game away from their opponents.
India, who beat arch-rivals Pakistan on Friday and take on qualifiers Bangladesh next Friday, top Group Two with four points.
Put into bat, West Indies opener Chris Gayle appeared to live a charmed life as Ravichandran Ashwin first dropped him in the slips before he had scored and Yuvraj Singh spilled a catch in the deep when the batsman was on 19.
The hard-hitting left-hander hit a couple of sixes but often struggled to find the meat of the bat. His nearly run-a-ball 34 stay ended after a mix-up with Marlon Samuels before the Indian spinners drove home the advantage.
Amit Mishra removed Samuels and Dwayne Bravo off successive deliveries to peg back West Indies before fellow spinner Ravindra Jadeja wrecked the middle order claiming the next three wickets, including two in the final over.