Jos Buttler smashed the league's second century of the season — 103 off 61 balls — to help Rajasthan post 5-217, the highest total of the season on Monday, but Kolkata looked like chasing it down at 4-178 entering the 17th over.
Then Chahal turned the momentum in Rajasthan's favour. Venkatesh Iyer was stumped, then three balls later captain Shreyas Iyer was trapped for 85 off 51 balls, and Shivam Mavi and Australian Pat Cummins were out to catches for ducks.
The 21st hat trick in IPL history, and the fifth by a Rajasthan bowler, gave Chahal 5-40.
However, Kolkata still has a shot. Umesh Yadav and Sheldon Jackson took 20 off the 18th over, and 7 off the 19th. They needed a gettable nine off the last five deliveries by left-arm pacer McCoy on debut, when Jackson mistimed a pull to short fine leg, and two balls later Yadav was bowled for 21 from nine.
Kolkata was all out for 210 with three balls to spare.
Rajasthan moved to second in the standings on run rate, while Kolkata's third straight loss put it at sixth.
Put in to bat, Buttler lit up Rajasthan's innings as he smacked nine fours and five sixes.
He raced to 50 off 29 balls, and his second hundred of this IPL came off 59 deliveries. He joined the likes of Chris Gayle (2011), Shane Watson (2018) and Hashim Amla (2017) as overseas batsmen with two hundreds in an IPL. Overall, it was his third IPL century for Rajasthan.
Buttler put on 97 runs off 58 balls for the first wicket with Devdutt Padikkal (24), and added 67 runs off 34 balls with Sanju Samson (38). He fell to Cummins late in the innings.
In reply for Kolkata', Australian Aaron Finch helped himself to 58 off 28 balls, including nine fours and two sixes, and resurrected their chase in a century stand with Shreyas Iyer.
Then Chahal's spin ruined it for them.
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