With India needing 34 runs off the last two overs, Karthik hammered 29 not out off just eight deliveries, his third six sealing India's third, and the most hard-earned, victory over Bangladesh in the tournament.
Put into bat, Bangladesh made 166-8, a total they owe largely to number three batsman Sabbir Rahman who made 77 after their top order collapsed.
Chasing 167 to win, India lost openers Shikhar Dhawan and number three Suresh Raina early but skipper Rohit Sharma batted comfortably to score 56 and keep India's chase on track.
He and KL Rahul (24) found it easy going against the Bangladesh bowlers until Sharma went for a fatal slog-sweep of Nazmul Islam, departing with three sixes and four boundaries to his name.
Mustafizur Rahman then bowled a masterly 18th over, when his first five balls at Vijay Shankar yielded only one run and Manish Pandey (28) perished off the sixth trying to accelerate the contest.
Next man in Karthik exploded immediately, hitting the first ball he faced for a six and plundering 22 runs off that over from Rubel Hossain to turn the match on its head.
Needing 12 runs off the last over from Soumya Sarkar, Shankar fell to the fifth delivery with India still five runs away.
Karthik nonchalantly hit the final ball for a six to trigger wild celebration by a young India team which had rested regular skipper Virat Kohli and stumper Mahendra Singh Dhoni, among others.
Interim captain Sharma had earlier opted to bowl after winning the toss, influenced partly by the fact that chasing teams had won five of the six matches prior to the final.
Bangladesh suffered early jolts and slumped to 68 for four midway through their innings after leg-spinner Yuzvendra Chahal (3-18) struck three blows, dismissing Tamim Iqbal, Soumya Sarkar and Mushfiqur Rahim.
Sabbir took the onus on himself to rescue Bangladesh and he got some support from Mahmudulla who made 21.
Even though wickets kept tumbling at the other end, Sabbir brought up his half-century off 37 balls to keep the scoreboard ticking.
Paceman Jaydev Unadkat struck twice in the penultimate over of the Bangladesh innings, dismissing Sabbir and Rubel Hossain.
Mehidy Hasan Miraz made 19 not out off seven balls to help Bangladesh cross the 150-mark.
(Reporting by Amlan Chakraborty in New Delhi; Editing by Christian Radnedge)
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