Chasing a tricky 241 for victory in the final, Australia slipped to 47-3 before the left-handed Head hit his second century of the tournament to steer the team home with seven overs to spare.
Head's knock and his marathon stand of 192 with Marnus Labuschagne, unbeaten on 58, ended India's dominant run of 10 unbeaten matches at the event.
Head fell after his 120-ball knock laced with 15 fours and four sixes before Glenn Maxwell hit the winning runs to trigger wild celebrations in the Aussie camp.
"Just thrilled to be a part of it," man-of-the-match Head told Star Sports.
"It's a lot better than seeing the World Cup on the couch at home (on his injury). I was a little bit nervous but Marnus played exceptionally well and soaked all the pressure."
India's chances of ending a global trophy drought since their 2013 Champions Trophy win went up in smoke once Head got going with Labuschagne.
Head's century was the seventh in a World Cup final and third by an Australian after Ricky Ponting (140 not out v India in 2003) and Adam Gilchrist (149 v Sri Lanka in 2007).
"The result hasn't gone our way and we know that we were not good enough on the day. But proud of the team," said India captain Rohit Sharma.
"With 240 on the board, we wanted early wickets but credit to Travis Head and Marnus. They put us completely out of the game and I thought the wicket got better to bat on under lights."
The bowlers set up victory for an Australian side that bounced back after two losses to win nine in a row as Mitchell Starc (3-55) and Pat Cummins (2-34) helped bowl out India for 240.
India hit back when Mohammed Shami shared the new ball with Jasprit Bumrah and struck on his second delivery to get David Warner caught behind for seven.
But it was Bumrah's double strike in quick succession that raised the roof as he had Mitchell Marsh caught behind for 15 and Steve Smith lbw for four.
Head stood firm with Labuschagne for company to thwart the Indian attack despite captain Rohit Sharma rotating his bowlers in a hunt for a breakthrough.
'Unbelievable' achievement
Head hit a match-winning century against New Zealand in the team's sixth league game and after a few low scores made an attacking 62 in the nervy three-wicket semi-final win over South Africa in Kolkata.
He proved to be India's nemesis a second time this year after his 163 proved decisive in Australia's World Test Championship triumph at The Oval in June.
Head reached his 100 in 95 balls and raised his bat to an applauding Australian dressing room.
"What we've achieved today is unbelievable," said Labuschagne.
"It's the best achievement I've ever been part of. India have been the team of the tournament, but you know if you play your best cricket, you have a chance. Our bowlers were sensational and Travis put on one hell of a display."
Warner said: "Our bowlers were fantastic, they set the tone from ball one. The fielding supported that."
Australia elected to field first and the players backed up Cummins' decision with disciplined bowling and impressive fielding.
Virat Kohli and KL Rahul hit 54 and 66 respectively after Rohit's attacking 47 but the ball dominated the bat on a slow, dry pitch.
Head took a stunning catch while running back from cover point to cut short Rohit's innings off-spinner Maxwell.
Cummins bowled Kohli, who ended as the leading batsman in the tournament with 765 runs, to silence the crowd of 92,453 fans, who like the home team in the middle had an unforgettable day.
