Kolkata won the toss and elected to bowl, removing Virender Sehwag for just seven in the fourth over.
He scored 115 not out off 55 balls - just the third century in the IPL this year and the first ever in a final.
His 129-run third-wicket partnership with Manan Vohra (67) ignited Punjab to an imposing total that gave Kolkata a victory target of exactly 200.
The Knight Riders' star man Robin Uthappa, who had scored almost 100 more total runs than any other batsman in this year's tournament, failed to have an impact on the final as Mitchell Johnson had him caught for just five.
Pandey motored towards his century and was ably assisted by Yusuf Pathan's 36 off just 22 balls as the Knight Riders kept just inside the required run rate.
Pandey then splayed a Karanveer Singh delivery to Bailey six runs short of his 100 and the advantage seemed to be with Punjab going into the final overs.
However, Piyush Chawla would be the hero for Kolkata, hooking a six into the crowd on the last ball of the penultimate over before carving the winning boundary in the final over.
After failing to come inside the top three in the first four years, Kolkata have now won of two of the last three IPLs. They are just the second team after Chennai Super Kings to win more than one title.