Bairstow's unbeaten 108 off 48 balls, with eight fours and nine sixes, see Punjab secure the win with eight balls to spare
A waterlogged outfield stopped play from starting on time and afternoon showers extinguished any hope of action at the Queen's Park Oval in Port of Spain.
Only 22 overs were bowled on Thursday before the rain arrived, West Indies moving to 62 for two after captain Jason Holder won the toss and batted.
More rain is forecast on Saturday, suggesting the final Test is headed for a draw.
India lead 2-0 and have already wrapped up the series.