The Australian allrounder helped lift Kings to 4-222 in their 20 overs with compatriot Cooper Connolly chipping in with a 14-ball 30, the pair hitting nine sixes between them.
On a tricky pitch, described by Kings skipper Shreyas Ayer as "tacky and slow", it looked sufficient. Yet the Royals reached their target with four balls and six wickets to spare, Donovan Ferreira ending it with a six to reach 4-228.
Teen star Vaibhav Sooryavanshi launched the response with an explosive 43 off 16 balls before rapid half-centuries from Yashasvi Jaiswal (51 off 27) and Ferreira (52 no off 26) took the visitors to victory.
The Ponting-coached Kings had won six matches, with the other washed off, before Tuesday's tie, and remain top of the table.
But bowlers Arshdeep Singh (68-1 off four overs) Lockie Ferguson (57-0 off four) and Marco Jansen (41-0 off 3.2) were all savaged as Royals hit 14 sixes.
Five of those came from Sooryavanshi, the remarkable 15-year-old who has set the cricket world alight.
The teen became the fastest to 400 runs in an IPL season, doing so in 167 balls, beating Andre Russell's record set in 2019 by 21 deliveries.
"I really enjoy batting with him. He is playing amazing, always happy seeing from the other end the way he is hitting the ball," said Jaiswal.
Put in to bat Kings began slowly but the second over, bowled by Nandre Burger, went for 21 and the batters were off and running.
Jofra Archer (1-40) sent back Priyansh Arya (29 off 11) but Connolly, after settling with two singles off five balls, hit successive sixes off Yash Raj Punja and one off Ravindra Jadeja only to be caught attempting to hit the former for another.
So well have Kings' top order performed this campaign Stoinis has mainly watched from the dugout. He had hitherto faced 27 balls scoring 48 in four innings for twice out.
Arriving after Prabhsimran Singh's 59 off 44 he unsurprisingly he took a few balls to get his eye in, scoring five from five.
Having had a "look" he smacked his next 17 balls for 57, including a brace of sixes off the disbelieving Archer, pulled over deep mid-wicket then cracked into the sightscreen in successive balls.
The West Australian took 24 off the final over, from Brijesh Sharma, and said with relish, "Enjoyed that. It has been a different season, watching the boys smack it all around the park. Nice to get a hit tonight."
The smile faded, however, as Sooryavanshi set about Arshdeep and Ferguson. The latter was having his first outing of the tournament having replaced Xavier Bartlett in the line-up, but saw Sooryavanshi whip a 142.5kph ball off his legs and over the fence, then cart his follow-up slower ball over mid-on.
Sooryavanshi was eventually caught off a skier, but Royals were well on their way by then.
Meanwhile, in the Pakistan Super League, Aaron Hardie took 3-24 as Peshawar Zalmi won the play-off with Islamabad United to reach Sunday's final.
Zalmi made 7-221 with Babar Azam scoring 103 off 59 balls, then bowled out Islamabad for 151.
