It was the fifth 10-wicket defeat for India in ODI cricket, and only the second at home.
Aaron Finch won the toss and put India in to bat.
Australia's first 10-wicket victory against India was fuelled by the third-highest partnership in ODIs between Warner and Finch for their team.
"I think I've always had the hunger and desire for runs".
"It was a great effort from our bowlers in the afternoon". India ended up with a below-par 255 on a decent batting surface after a middle-order failure.
He and Finch started the chase in belligerent fashion, putting on 84 runs in the first powerplay.
He has a technique and mental fortitude that probably should have earned him a century by now, instead of just his eight half-tons and a personal best of 74 against Pakistan at SuperSport Park in February 2013.
Australia's progress might have been less relentless had Shardul Thakur's appeal for a leg-side catch when Warner had made five been upheld - but the batsman's review proved successful.
India's bowlers failed to get any purchase, and the dewy conditions in Mumbai didn't help their cause as Australia galloped to 200 in the 31st over.
Finch completed his 16th century in the format, his unbeaten innings of 110 studded with 13 fours and two sixes.
Virat Kohli is one of the most fierce competitors on the field but off the field, he is known to be quite a gentleman.
One of the most topsy-turvy Tests imaginable saw Philander bag a sensational, quickfire haul of five for 15 as the Baggy Greens were routed for 47 (they were a more jaw-dropping 21 for nine at one point) in their second turn at the crease, the Graeme Smith-led Proteas going on to win by eight wickets.
Pat Cummins struck in the next over to dismiss Shikhar Dhawan.
Ravindra Jadeja (25 from 32) and Rishabh Pant (28 from 33) restored some momentum to the innings with a stand of 49, but both departed in quick succession and it was left to Kuldeep Yadav (17) to steer them above the 250 mark. "I knew that in a few overs we would have gone for it, like I started going for it and started to accelerate three-four overs before KL got out". In phases, we were too respectful against their bowlers, and didn't take the game by the scruff of the neck. I think people need to relax and not panic with one game, I'm allowed to experiment a little bit and fail as well at times.
"It's become a very exciting feature of any Test series, so we are absolutely open to play a day-night Test", Kohli told reporters in Mumbai.