Earlier Bairstow smashed New Zealand quick Trent Boult for two boundaries in the first over to signal his intent as he helped Punjab to 48-1 in the first six overs.
But left-handed Hetmyer smashed three fours and two sixes in his 16-ball blitz to ensure the Royals bounced back from two losses and stayed third in the 10-team table.
He lost his opening partner Shikhar Dhawan to a stunning catch from Buttler, who backpedalled from mid-on and then timed his jump to pluck the ball one-handed. Ravichandran Ashwin was the bowler.
Bairstow put on key partnerships including a 42-run second-wicket stand with Bhanuka Rajapaksa before he fell lbw to Yuzvendra Chahal, who returned figures of 3-28.
Wicketkeeper-batter Jitesh Sharma kept up the attack and found Liam Livingstone's company as the duo added 50 off 26 balls for the fifth wicket.
Livingstone was bowled by Prasidh Krishna for 22 off 14 balls, but Sharma finished with a flourish as he hit a six and two fours in a 16-run 20th over.
The effort was not enough as the Royals won their first match chasing this season.