Chasing a tricky 145 on a sluggish pitch, Punjab depended on Curran's unbeaten 63 to achieve their target with seven balls to spare in Rajasthan's adopted home ground in Guwahati.
Left-handed Curran put on a key 63-run stand with Jitesh Sharma, who made 22, and despite losing his partner in the 16th over steered the team home to lift them from the bottom of the 10-team table.
Curran, who took over as Punjab captain midway this season after an injury to regular skipper Shikhar Dhawan, and teammate Jonny Bairstow will depart for England on Thursday in preparation for the T20 World Cup in June.
"Me and Jonny are leaving tomorrow," player of the match Curran said.
"Obviously nice. Have thoroughly enjoyed the season, have enjoyed captaining. Atmosphere around the group has been really good. Excited about the World Cup."
Most of the England players in IPL including Rajasthan's Jos Buttler have already left home ahead of four T20 matches against Pakistan as a tune-up for the World Cup in the West Indies and the United States.
Impact substitute Ashutosh Sharma smashed 17 and the winning run to hand Rajasthan their fourth successive defeat in this edition of the T20 tournament.
"To be very honest, we have to sit back and accept that we are going through some failures," Rajasthan skipper Sanju Samson said.
"You have to find out what's not working well as a team. When you are getting to the business end, we need someone to raise their finger up and say 'I am going to win the game for the team'. We have the players who can do that."
The loss for Rajasthan hurt their chances of a top-two finish in the league stage a day after they confirmed their play-off spot.
Rajasthan, who stay in second place with one game in hand, and table-toppers Kolkata Knight Riders are the only two teams to have booked their play-off berth.
The top four teams make the play-offs, but the first and second-placed sides have the advantage of two chances to qualify for the final on May 26 in Chennai.
Punjab, who had already bowed out of the play-off race, kept the opposition down to 144-9 despite a 34-ball 48 by Riyan Parag.
Left-arm quick Curran bowled Yashasvi Jaiswal, for four, in the first over and then Samson departed for 18 off Nathan Ellis after a slow start.
Parag found Ravichandran Ashwin, who hit 28 off 19 balls, for company and the two attempted to push the scoring in their partnership of 50.
The rest of the batting faltered against a disciplined attack as Curran, fellow quick Harshal Patel and spinner Rahul Chahal took two wickets each.
