Sharma, who batted sensibly after India had slumped to 104 for four following the loss of three early wickets and opener Shikhar Dhawan for 51, made 68 not out in 75 balls as the world champions wrapped up victory with more than five overs to spare.
West Indies, batting first on a low bouncing pitch that suited spinners, could only manage 214 for nine from their 50 overs with Ramnaresh Sarwan (56) and Marlon Samuels (55) combining for more than half the team’s score.
Only three other batsmen made double figures and none scored more than 22. Sarwan and Samuels put on 82 for the fourth wicket but when they departed the tourists were able to stem the run flow and pick up regular wickets.
Harbhajan Singh was the pick of the Indian bowlers, capturing three for 32 from 10 overs.
India, playing their first ODI since winning the World Cup on home soil in April, were missing captain Mahendra Singh Dhoni and master batsman Sachin Tendulkar and made a shaky start to their reply.
They lost three wickets in the first 16 overs and the game was still in the balance when Dhawan threw his wicket away just after reaching his half-century when he holed out in the deep.
But Sharma knuckled down and shared a vital 80-run stand with Suresh Raina that put India back on course.
Raina departed for 43 and although India lost a sixth wicket they got home safely, Harbhajan sealing the win with a six.
Sharma, who also hit a six and three boundaries, was named man of the match.
The second of the five matches is scheduled for Wednesday. The teams also play a three-match test series starting at the end of June.