India needed just five off the last over but the tension rose when Dhawan lifted Allen straight to Pollard at long-on and a tie looked to be on the cards when Pandey could only prod the ball to the right of the bowler, who was devastated after failing to pull off what should have been a run out.
West Indies, after electing to bat, made 181 on the loss of three wickets.
Earlier Pooran put on a key 87-run unbeaten stand with Darren Bravo, who made 43, to give the tourists a competitive total.
However, the likes of Shikhar Dhawan, Rishabh Pant and KL Rahul have been scratchy throughout the series and have not been able to muster runs from their willows.
The Indian innings started on a disappointing note for Rohit Sharma's fans as the skipper fell after scoring a boundary, caught by his opposite number Carlos Brathwaite off Kemo Paul's bowling. Pooran blasted four fours and as many sixes in his 25-ball unbeaten innings.
Chahal got Hope on his very first delivery and then cut short Hetmyer's brisk knock to check the opposition surge.
Chahal, who got a lot of turn off the wicket, was the most successful Indian bowler with 2 for 28 from his four overs, but faced the heat towards the end when Pooran and Bravo went on the attack.Washington Sundar, who took the wicket of Denesh Ramdin (15), was the other wicket-taker. We lost two early wickets, and I knew I had to stay at the crease.
It will be interesting to see whether the team management tweaks the spin department comprising leg-spinner Yuzvendra Chahal and left-armer Krunal Pandya, with Sundar and Nadeem still warming the bench.
The result caps off a complete domination by India, who had won the Tests 2-0 and then the one-day worldwide series 3-1 against the visiting West Indians.
Opting to bat, West Indies retained the same playing XI in the last match on the tour, an inconsequential one, after India sealed the three-match series 2-0 in Lucknow.