Bairstow's unbeaten 108 off 48 balls, with eight fours and nine sixes, see Punjab secure the win with eight balls to spare
Bangladesh reached 200-3 in the 60th over on the final day of the match, with Haque on 74 and Liton Das on 56.
Sri Lankan spinners Rangana Herath and Lakshan Sandakan, each of whom picked up a wicket late on Saturday, are leading the bowling attack.