Hope, who hit a hundred in the second game of the series, batted the entire to be unbeaten on 102 from 50 balls as the Windies posted an imposing total of 247-3 after being sent in to bat at the toss.
The visiting skipper received good support through the middle-order, with Shimron Hetmyer striking 33, Rovman Powell 24 while the innings received a late boost when Sherfane Rutherford smashed an unbeaten 49 from just 26 deliveries.
India initially made a good fist of reaching the 248-run target with opener Sanju Samson hitting 56, but the home team suffered a serious collapse to be 130-5 in the 12th over.
Paul took the key wicket Samson and the seamer went on a bit of a roll towards the end as the hosts looked for quick runs to finish with figures of 5-41 in his four overs as India were 203 all out.
The fifth and deciding encounter of the series will take place in Bengaluru on Saturday.
India top-order batter Ishan Kishan has been ruled out of the second T20I in New Zealand due to a groin strain, with teenage star Vaibhav Sooryavanshi poised to return to the starting XI.
India earned their highest-ever T20I run chase when they successfully chalked off New Zealand's 225/4 with four wickets remaining in a run-soaked encounter in Christchurch on Thursday night.
Darren Sammy believes West Indies have taken a significant step forward after holding India to a 2-2 T20I series draw, with the head coach encouraged by the return of his team's confidence and trademark swagger.
Less than eight overs were bowled in the fifth and final T20I between India and West Indies as heavy rain fell in Bengaluru.
