De Kock hit 140 off 70 balls laced with 10 sixes after Lucknow, one of the two newcomers alongside already-qualified Gujarat Titans, in the expanded 10-team Indian Premier League, elected to bat first.
Rahul made 68 off 51 deliveries but De Kock was unforgiving on the bowlers as he smashed New Zealand quick Tim Southee for four hit over the fence in a 27-run 19th over and then hit Andre Russell for four fours in the 20th.
The South African wicketkeeper-batsman survived a dropped catch on 12 and went on to score his second IPL century and dropped to the ground in an emotional celebration.
He hit this IPL's best individual score to better the 116 by Rajasthan Royals' batsman Jos Buttler.
The left-right batting pair of De Kock and Rahul also surpassed the previous best first innings partnership of 185 between Jonny Bairstow and David Warner for Sunrisers Hyderabad in 2019.
It's the third best partnership for any wicket after two top stands by Virat Kohli and AB de Villiers of 229 (2016) and an unbeaten 215 (2015) -- both for the second wicket for Royal Challengers Bangalore.
Two-time champions Kolkata lost openers Venkatesh Iyer, caught behind for nought with De Kock taking a superb diving catch, and Abhijeet Tomar, for four, to left-arm fast bowler Mohsin Khan who returned figures of 3-20.
Kolkata lost wickets and hope after contributions by Nitish Rana, who made a 22-ball 42, skipper Shreyas Iyer, who fell for 50 off 29 balls and Sam Billings' 36.
But Singh and Sunil Narine, who made a seven-ball 21, took Lucknow by surprise with a partnership of 58 off 19 balls.
Stoinis held his nerve to get Rinku and then bowled Umesh Yadav on the final ball to trigger wild celebrations in the Lucknow camp.