South Africa required 16 runs to win after Heinrich Klaasen had put them firmly on course in reply to India’s 176 for seven by smashing 52 from 27 balls including five sixes.
David Miller hit the first ball, a full toss from Hardik Pandya, high down the ground but Suryakumar Yadav raced around the long-off boundary, knocked the ball into the air, toppled over the ropes and stepped back to complete a stunning catch.
Two boundaries from the final two balls would still have led to a Super Over with the scores tied but Kagiso Rabada was caught off the fifth and the match was over as jubilant Indian supporters swarmed onto the ground.
After Rohit Sharma had won the toss and opted to bat, Virat Kohli finally recaptured his best form with 76 from 59 balls.
After scoring only 75 runs in seven knocks at the tournament, Kohli first anchored the innings after India had lost their top three wickets cheaply before accelerating.
Kohli signalled his intent by striking three boundaries in the opening over from paceman Marco Jansen but South Africa struck back immediately through left-arm spinner Keshav Maharaj.
Maharaj dismissed Rohit for nine and had Rishabh Pant caught by Quinton de Kock for a duck off another mistimed sweep.
Suryakumar (three) lofted paceman Rabada to the square leg boundary where Klaasen took a comfortable catch and after the powerplay, India were reeling on 45 for three.
Left-hander Axar Patel lofted the first six of the match over mid-wicket in the eighth over as India sought to accelerate, reaching 75 for three at the halfway stage.
Axar was run out for 47 when De Kock threw down the stumps at the bowler’s end with the batter centimetres short of his ground.
Shivam Dube slapped a six and a four and Kohli brought up his half-century from 44 deliveries.
Kohli was now in full flow, smashing Jansen for six before being caught by Rabada trying another hit over the boundary.
Bumrah Strikes
Jasprit Bumrah, India’s strike bowler, almost inevitably struck in his opening over when he clean-bowled Reeza Hendricks.
De Kock, however, went on to the attack, taking a four and a six off Kuldeep Yadav and he kept South Africa up with the required run rate to reach 39 from 31 balls when he swung left-arm paceman Arshdeep Singh straight to Kuldeep at fine leg.
Klaasen’s pugnacious innings put his team within sight of victory with 22 runs required off 18 balls when Rohit turned to Bumrah to bowl his final overs.
Bumrah responded by bowling Jansen for two while conceding just two runs to finish with two for 18 in another magnificent spell of bowling.
"I tried to keep calm," said Bumrah, who was named player of the tournament.
"We play sport for the big stages. On the big day you have to give more, throughout the tournament I felt very clear and calm."
Man-of-the-match Kohli retired from T20 Internationals soon after the victory as India became the first side to win the trophy after going unbeaten through the tournament.
If India players were in tears after winning their first global title since the 2013 Champions Trophy, their South African counterparts looked desolate after another heartbreak.
"Gutted. It will take some time for us to reflect on this," South Africa skipper Aiden Markram said.
"We've had a great campaign but for the time being, this hurts.
"I am so proud of all my players and everyone involved in this team."
