The match had ended in a tie after Delhi, chasing 186 for victory, finished their 20 overs on 185 for six, with Prithvi Shaw hitting 99 at Delhi's Feroz Shah Kotla stadium.
Rabada then returned to bowl two perfect yorkers in the Super Over - the first of this edition of the world's most popular Twenty20 tournament.
Kolkata batsman Andre Russell, who had smashed 62 off 28 deliveries in his team's 185-8, was at the receiving end of Rabada's accuracy.
The South African paceman rattled Russell's stumps on the third ball, with Kolkata managing just seven runs in their six deliveries, in reply to Delhi's 10.
"Obviously lots of nerves... You have to keep your wits about you and be really clear on what you have to bowl," said the 23-year-old Rabada.
"I just went with yorkers and it worked in the end. I was clear and backed my plan."
The T20 extravaganza is tentatively scheduled to end on May 12 with eight franchises battling it out in season 12 of the cash-rich league.