India dismissed Australia for 291 with two balls remaining in an extended penultimate session on day five.
Set 323 runs to win, Australia's lower-order provided gritty resistance to edge toward the target, but the hosts were bowled out after 119.5 overs in the second innings. India shared the wickets around with Ravichandran Ashwin, Jasprit Bumrit and Mohammed Shami claiming three each.
Pant equaled a world record for wicketkeepers with 11 catches for the test. Cheteshwar Pujara was man-of-the-match after scores of 123 and 71. The second test in the four-match series starts Friday in Perth.
The match appeared over when Pat Cummins fell at 259-9, but Nathan Lyon and Josh Hazlewood defied India's flagging attack with a last-wicket partnership of 32 runs. Hazlewood (13) was caught at first slip off Ashwin, sparking jubilation from India's players.
Lyon finished 38 not out from 47 deliveries.
Australia's slim chances appeared to nosedive when captain Tim Paine was dismissed on his first delivery after lunch, throwing his wicket away with a miscued pulled shot off Bumrah. He made 41 after aggressive batting in the morning session and required treatment on his troubled right index finger.
Cummins (28) and an attacking Mitchell Starc (28) combined to frustrate India with an eighth-wicket partnership of 41 runs. Starc's breezy knock ended when he edged a wide Shami delivery to be caught behind.
Cummins batted with the concentration and application often lacking from Australia's top-order batsmen, with his 121 balls faced the second most in the innings. His innings ended when he edged Bumrah to captain Virat Kohli, who glared at the crowd after snaring the first slip catch.
Australia's main batting hopes —Shaun Marsh and Travis Head — were dismissed before lunch on the last day. Head made 14 and the under-pressure Marsh fell for 60 from 166 balls late in the session.
Play started 30 minutes early after a rain-interrupted day three, and a minimum of 98 overs were to be bowled. Chasing a record target at the Adelaide Oval, Australia resumed at 104-4 with Marsh on 31 and Head on 11.
Each run was applauded by the sparse crowd of 10,868 but Australia struggled against tight bowling, with the first boundary struck 40 minutes into the day's play.
Ishant Sharma broke through with a sharp bouncer, hitting the shoulder of Head's bat and getting an edge to gully.
Marsh reached a half-century with a pull to the boundary off Ashwin, his first time passing 50 in the fourth innings of a test.
India won its first test in Adelaide since 2003, when it beat a Steve Waugh-led Australia by four wickets. India's last test match win in Australia was during the 2007-08 series, and it has never won a test series Down Under.