Set a target of 252 at the Sir Vivian Richards Stadium in Antigua to prevent the visitors taking an unbeatable 2-0 series lead, the hosts were on course with Jason Mohammed and Rovman Powell in full flow in a 54-run sixth-wicket partnership.
But then India's frontline spinners Kuldeep Yadav and Ravichandran Ashwin swung the match decisively in their team's favour.
From a promising position of 141 for five in the 33rd over, the Caribbean side crashed to 158 all out within the next five overs.
Yadav and Ashwin took three wickets each in engineering the capitulation that was finished off by Kedar Jadhav as he yorked last man Kesrick Williams with his first ball of the match.
Powell had looked dangerous in getting to 30 before a top-edged sweep off Yadav resulted in the pivotal dismissal via a catch at deep backward square leg.
Mohammed, who managed a top score of 40, became another victim of the left-arm wrist-spinner, trapped leg-before to effectively end the contest.
Earlier it was Jadhav who triggered a late run explosion in partnership with Mahendra Singh Dhoni as India reached 251 for four batting first on a sluggish pitch affected by prolonged, torrential rains the day before.
Dhoni top-scored with an unbeaten 78 off 79 balls and was named man of the match, but the real impetus came from Jadhav, who blazed 40 off 26 balls in an unbroken fifth-wicket stand of 81 runs off just 46 deliveries.
Ajinkya Rahane held the innings together after the loss of early wickets, the opener extending his excellent run of form in the series with 72, his third consecutive score over 50.
Comprehensively beaten by 105 runs in the second match in Trinidad five days earlier following a rain-ruined no-result to start the series, West Indies took early advantage of the pitch conditions.
A. Rahane c Bishoo b Cummins 72
S. Dhawan c Chase b Cummins 2
V. Kohli c K. Hope b Holder 11
Y. Singh lbw Bishoo 39
M.S. Dhoni not out 78
K. Jadhav not out 40
Extras (lb-1, w-8) 9
Total (4 wkts, 50 overs) 251
Fall of wickets: 1-11 (Dhawan), 2-34 (Kohli), 3-100 (Yuvraj Singh), 4-170 (Rahane)
Did not bat: H. Pandya, K. Yadav, R. Ashwin, B. Kumar, U. Yadav
Bowling: M. Cummins 10-0-56-2 (2w), J. Holder 10-1-53-1 (2w), K. Williams 10-0-69-0 (1w), A. Nurse 10-0-34-0 (1w), D. Bishoo 10-0-38-1 (2w)
E. Lewis b U. Yadav 2
K. Hope c Jadhav b Pandya 19
S. Hope c and b Pandya 24
R. Chase b K. Yadav 2
J. Mohammed lbw K. Yadav 40
J. Holder stpd wkpr Dhoni b Ashwin 6
R. Powell c Pandya b K. Yadav 30
A. Nurse c U. Yadav b Ashwin 6
D. Bishoo not out 4
M. Cummins lbw Ashwin 1
K. Williams b Jadhav 1
Extras (lb-6, w-17) 23
Total (all out, 38.1 overs) 158
Fall of wickets: 1-9 (Lewis), 2-54 (K. Hope), 3-58 (Chase), 4-69 (S. Hope), 5-87 (Holder), 6-141 (Powell), 7-148 (Nurse), 8-156 (Mohammed), 9-157 (Cummins)
Bowling: B. Kumar 5-0-19-0, U. Yadav 7-0-32-1 (5w), H. Pandya 6-0-32-2 (3w), K. Yadav 10-1-41-3, R. Ashwin 10-1-28-3 (3w), K. Jadhav 0.1-0-0-1
Result: India won by 93 runs