The nail-biting four-wicket in Dharamsala win gave Rajasthan, last season's runners-up, hope of qualifying for the knockout phase although their fate depends on other matches this weekend.
Curran, who cost Punjab a hefty $2.23 million, fell one run short of what would have been only his second 50 of the tournament as Punjab put on 187-5 with Navdeep Saini taking 3-40 for Rajasthan.
This was after Punjab were in trouble at 50-4 after 6.3 overs when Saini sent Liam Livingstone's off stump flying, the English big-hitter managing just nine and captain Shikhar Dhawan 17.
Curran's valiant 49 came off 31 balls with four fours and two sixes included vital partnerships with Jitesh Sharma (44) and Shahrukh Khan (41) that rescued Punjab's innings.
In reply for the Royals, Curran's England teammate Jos Buttler fell for his latest duck of his lacklustre season, trapped leg-before to a vicious inswinger by Kagiso Rabada.
But Jaiswal, the IPL's second-highest run scorer this year behind Faf du Plessis, and who last week hit a 13-ball 50, continued in his stellar form with yet another half-century, off 36 balls.
Tense finale
Rajasthan captain Sanju Samson departed for just two to Rahul Chahar but rock-solid Devdutt Padikkal made 51 off 30 before falling to a stunning catch on the circle by Harpreet Brar off Arshdeep Singh.
Hetmyer almost saw the Royals home in a tense final phase that saw the West Indian exchange sharp words with Curran when he was given out caught behind, only to be reprieved on review.
"It is nice to have someone saying something, giving me a little bit more confidence and a little bit more energy to execute," Hetmyer said afterwards, adding that what Curran said to him "is not anything I can disclose".
Curran, who leaked 46 runs in the match, finally got his man and his only wicket in the 19th over, with Hetmyer finally departing for 46 off 28 after a brilliant catch by Dhawan at cover.
But Hetmyer and Riyan Parag (20) had done enough, with impact player Dhruv Jurel (10) finishing the job by clubbing a six back over Chahar's head with two balls remaining.
The win in their final group-stage game by Rajasthan puts them in fifth position with 14 points.
Only the top four proceed, and whether Rajasthan make it depends on other teams, primarily Royal Challengers Bangalore and Mumbai Indians, currently fourth and sixth respectively.
Defending champions Gujarat Titans are top of the table ahead of second-placed Chennai Super Kings. Punjab though are out, as are Delhi Capitals and Sunrisers Hyderabad.
"I think that in all areas we could have done better. Sometimes the batting clicked and sometimes the bowling clicking," said Dhawan.
"I am sure this season boys must have learned lots of things. As a leader I learned lots of things," he said.
