Bairstow's unbeaten 108 off 48 balls, with eight fours and nine sixes, see Punjab secure the win with eight balls to spare
The result meant India won the four-Test series 2-0 but surrendered top spot in the International Cricket Council (ICC) rankings to Pakistan, less than a week after taking over the number one position from Australia.
India had needed to win in Port of Spain to keep the top ranking but only 22 overs were completed on Thursday, with play completely washed out over the following four days at Queen's Park Oval.
The fixture, played in Trinidad's rainy season, became the third-shortest non-abandoned Test ever.
India and West Indies now move to Florida for two Twenty20 matches this weekend.