Scorecard
The little-known Bisla smashed five sixes and eight fours in his quickfire 48-ball 89 as Kolkata surpassed Chennai's total of 190-3 with two balls to spare in the Twenty20 final before a sizeable crowd at the Chidambaram Stadium.
Kolkata needed nine runs to win off the last over bowled by West Indies seamer Dwayne Bravo, but Manoj Tiwary (nine not out) smashed the third and fourth deliveries for fours to spark celebrations in his team's camp.
Kallis made a crucial 49-ball 69 with the help of one six and seven fours despite suffering a hamstring injury in the later part of his knock.
Man of the Match Bisla and Kallis put on 136 runs for the second wicket in 13.4 overs before Tiwary and Bangladesh all-rounder Shakib Al Hasan (11 not out) completed the win to deny two-time champions Chennai a hat-trick.
Australian paceman Ben Hilfenhaus bagged two wickets for Chennai, who were facing elimination at one stage during the tournament.
Chennai were lucky to figure in the playoffs, having qualified on better net run-rate than that of Royal Challengers Bangalore after both the teams finished their league engagements with 17 points.
The final was also watched by Pakistan's cricket chief Zaka Ashraf. Pakistani players, who are usually a major draw in India, have been kept out of the tournament after the first edition, reportedly due to security fears.
Suresh Raina earlier cracked a 38-ball 73 with five sixes to help Chennai set a stiff target. Australian Michael Hussey (54), who turned 37 on Sunday, and Murali Vijay (42) were the other main scorers.
Chennai made a brisk start after electing to bat as Hussey put on 87 for the opening wicket with Vijay and then 73 for the second wicket with Raina.
Raina raced to his half-century off just 27 balls before falling off the last ball of the innings, caught by Australian Brett Lee in the deep off Shakib.
Kolkata got $2 million for the win while runners-up Chennai bagged $1.5 million.
Kolkata, Chennai and the Delhi Daredevils qualified for the Champions League to be held in India in October between the top domestic Twenty20 teams from around the world.
Brief scores:
Chennai Super Kings 190-3 in 20 overs (S. Raina 73, M. Hussey 54, M. Vijay 42) lost to Kolkata Knight Riders 192-5 in 19.4 overs (M. Bisla 89, J. Kallis 69; B. Hilfenhaus 2-25) by five wickets.