Chasing 270 for victory, Kohli (117) and Sharma (90 not out) batted with a lot of maturity and added 163 for the fourth wicket to give India a 2-0 lead in the five-match ODI series against the tourists.
Man-of-the-match Kohli hit 14 boundaries before edging a catch behind with India still 23 away from the win.
But Sharma, who brought up his second consecutive half-century in the series, kept his calm to take India home with 11 balls to spare.
India lost Parthiv Patel and Gautam Gambhir early but Kohli and captain Virender Sehwag (26) added 55 for the third wicket to steady the ship.
The visitors were still in the hunt after Sehwag departed trying to clear off-spinner Marlon Samuels over the long-off boundary and India were reduced to 84-3.
But the stand between Kohli and Sharma put paid to their hopes of leveling the series.
Earlier, tailender Ravi Rampaul blasted his way to an unbeaten 86 from the No. 10 position to deny India's hopes of restricting West Indies to a far lower total.
The left-hander hit six fours and six sixes in his 66-ball innings to lift West Indies to 269 for nine.
Rampaul, whose highest score in ODIs before this match was an unbeaten 26, added 99 in 14 overs for the unbroken final wicket with Kemar Roach (24 not out).
Pace duo Umesh Yadav (3-38) and R. Vinay Kumar (2-43) picked up the first five wickets to fall, reducing the visitors to 63-5, before opener Lendl Simmons (78) and Kieron Pollard (35) stemmed the rot with a 56-run stand for the sixth wicket.
Simmons anchored the innings with his patient 102-ball knock but was run out with West Indies on 170 and with just the last pair left to bat.
The third ODI will be played in Ahmedabad on Monday.
