The 2007 champions restricted their big-hitting West Indian opponents 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 opposition.
Kohli once again displayed a silken touch, scoring fluently to underline his growing stature as one of the most promising batsmen in the game, and he got good support from Sharma who batted until the end.
Needing five runs to win off 12 balls with eight wickets in hand, India's approach to victory was rather slow but their win was never in doubt even after they lost a struggling Yuvraj Singh in the final over.
India, who beat 2009 champions Pakistan on Friday and take on qualifiers Bangladesh next Friday, top Group Two with four points.
Captain Mahendra Singh Dhoni was particularly happy that India's previously misfiring bowling attack finally struck form.
"I'm really happy how the spinners are bowling," Dhoni told reporters.
"Yes, there is a bit of help for them but at the same time you have to execute your plans well, especially when you've some of the big hitters in the opposition.
"I was really happy that our spinners, along with the part-timers and fast bowlers, have done really well but there is still a long way to go in this tournament."
Put into bat, West Indies opener Chris Gayle appeared to live a charmed life as first Ravichandran Ashwin dropped him in the slips before he had scored and Yuvraj Singh then 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 innings ended on 34 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 the West Indies before fellow spinner Ravindra Jadeja wrecked the middle order, claiming the next three wickets, including two in the final over.
West Indies skipper Darren Sammy said his side, packed with big-hitters, never got any momentum going.
"We did not respond well to their spinners," Sammy said.
"Round one goes to India but we've a strong feeling that we'd meet again (in the tournament). We're looking forward to that.
"I think credit must go to the opponent bowlers. (Bhuvneshwar) Kumar swung the ball in and out and he bowled good areas.
"They were bowling to the two most dangerous openers (Gayle and Dwayne Smith) in this game (but) they kept them quiet.
"There is no need to panic, it's just one wrong. We've two more games left in the tournament and we back ourselves to win them," added Sammy, whose team face hosts Bangladesh on Tuesday.
- Sports & Recreation
- Mahendra Singh Dhoni
- Yuvraj Singh