Jadhav, whose unbeaten 87-ball 81 was laced with nine fours and a six, was supported by Mahendra Singh Dhoni (59 not out off 72 balls) as India took a 1-0 lead in the five match series, which is their last before the World Cup in May.
Wearing their brand new jerseys, the men in blue were asked to chase a modest 237 by the visitors after the home team bowlers did a brilliant job to restrict them to 236/7.
Dhoni who hit the winning runs, a four off Marcus Stoinis over the infield, with 10 balls to spare joined hands with Jadhav for an unbroken 141-run stand for the fifth wicket.
Dhoni came in to bat when India were in a spot at 99/4 with skipper Virat Kohli (44 off 45 balls) and vice-captain Rohit Sharma (37 off 66 balls) back in the hut.
Dhoni's innings was studded with six fours and a six but it was Jadhav who guided India past the finishing line. Jadhav scored his fifth half century, while Dhoni brought up his 71st in his 339th ODI, playing through cramps towards the later part of the match.
The visitors started well, shaving off the Indian top order under 100 runs by the 24th over. Nathan Coulter-Nile (2/46 in 10 overs) and Adam Zampa (2/49 in 10 overs) caused the damage.
Rohit and Kohli had stitched together a partnership of 76 for the second wicket but just when they were looking threatening, Kohli was trapped in front by Zampa. The umpire initially turned down the appeal, but reviews showed the ball had pitched in line and was hitting the wickets.
Rohit was removed by Coulter-Nile with a cross seam delivery that took the leading edge and went to mid-off for skipper Aaron Finch to grab the opportunity.
Ambati Rayudu (13) was caught behind off Zampa as India looked in trouble. From there on, Dhoni and Jadhav paced the chase to perfection with the former playing second fiddle and the latter taking the charge.
Earlier, Indian bowlers once again underlined their collective might by not allowing the Aussies to ever get going.
Pacer Mohammed Shami led the attack with figures of 2/44 in 10 overs as chinaman bowler Kuldeep Yadav (2/46 in 10 overs) Ravindra Jadeja (0/33 in 10 overs) and Jadhav (1/31) made up for a rare off-day by Jasprit Bumrah (2/60 in 10 overs).
Usman Khawaja (50) and Glenn Maxwell (40) scored the runs for the visitors but never looked in full flow as the Indian attack displayed its versatility.
Shami was on song from start to finish as he bowled a brilliant wicketless first spell, which was seen off by Stoinis (37) and Khawaja who added 87 runs for the second wicket after the early dismissal of skipper Aaron Finch for a duck.
The Powerplay yielded just 38 runs with Stoinis and Khawaja then changing gears off Vijay Shankar's bowling.
In the next five overs, Australia scored 33 runs and looked like regaining ground when Jadhav (1/31 from 7 overs) got a breakthrough with a half-tracker that Stoinis looked to whip away through the mid-wicket region.
Khawaja completed his sixth half-century in ODIs but couldn't continue, holing out to Shankar at deep mid-wicket boundary off Kuldeep.
Maxwell joined Peter Handscomb at 97/3 and the pair stemmed the rot somewhat.
Kuldeep then removed Handscomb with a tossed up delivery outside off stump that beat the batsman and Dhoni completed an easy stumping.
India then took the game by the scruff of the neck in the second Powerplay with Jadhav and Jadeja checking the run flow.
Shami castled Maxwell and debutant Ashton Turner (21) to leave the Aussies staring at a sub-200 score. But Alex Carey (36 off 37 not out) and Nathan Coulter-Nile (28 off 27) added 62 runs for the seventh wicket.
Brief scores: Australia 236/7 (Usman Khawaja 50, Glenn Maxwell 40; Mohammed Shami (2/44), Kuldeep Yadav (2/46) and Jasprit Bumrah (2/60); India 240/4 (Kedar Jadhav 81 not out, MS Dhoni 59 not out; Nathan Coulter-Nile 2/46, Adam Zampa 2/49)