Rajasthan posted 155-9 as their batting faltered after 15-year-old Vaibhav Sooryavanshi top-scored with 46 at Kolkata's Eden Gardens.
Kolkata, who had five defeats and one point from a rained-off game coming into the match, slipped to 5-2 and 85-6 before the left-handed Rinku steered the team home with two balls to spare.
Rinku smacked five fours and two sixes in his 34-ball knock and put on an unbeaten 76-run stand with number eight Anukul Roy, who made 29, to steer the team home.
The left-handed Roy kept calm with his senior partner as he hit a six off Jofra Archer in the 19th over and Kolkata needed nine from the final six balls.
Rinku, who was dropped on eight by Nandre Burger, hit two fours and a winning six as he raised his fifty to bring cheer to the dugout and home crowd.
England pace bowler Archer had Tim Seifert bowled for a duck on the first ball of the chase and South Africa's Burger dismissed skipper Ajinkya Rahane for nought to make it two down.
Kolkata's big-buy Cameron Green looked fluent in his 27 off 13 balls but fell stumped off Ravi Bishnoi with wicketkeeper Dhruv Jurel making a wide collection and a back flip onto the stumps.
Wickets kept tumbling with Ravindra Jadeja claiming two but Rinku made the most of his early reprieve to get three-time champions Kolkata over the line.
Earlier, Rajasthan started strongly with openers Yashasvi Jaiswal, who made 39, and T20 sensation Sooryavanshi, who bounced back from a golden duck in the previous game to hammer six fours and two sixes in his 28-ball knock, putting on 81 for the first wicket.
Spinner Varun Chakravarthy took down the left-handed Sooryavanshi and returned figures of 3-14 to trigger a collapse.
Fast bowler Kartik Tyagi took three key wickets and Sunil Narine claimed 2-26.