Harris reaches his second Test 50 with a single off Jasprit Bumrah then smacked three fours in one Yadav over.
Paine acknowledges that not having world-class batsmen was hindering their chances of competing with teams like India, ranked number one in the world.
It leaves India zeroing in on a first-ever series win in Australia with just the Sydney Test next week to go.
The overwhelming victory gave New Zealand the series 1-0 after the first Test in Wellington was drawn.
India started with a 346-run lead and with five wickets in hand on a wearing pitch becoming increasingly difficult to bat on.
Arthur stormed into third umpire Joel Wilson's room and questioned him.
Paine said he was not annoyed "in the slightest" by Kohli looking straight past him at the handshake.
Finch's right index finger suffered a nasty blow when during the second test in Perth, forcing him to retire hurt.
The hosts led by 43 after both teams' first innings despite a superb Virat Kohli century, but found batting incredibly tough in their second innings.
The imperious Kohli was unbeaten on 82 and shaping as the major stumbling block for Australia.
Southee claimed three wickets in his first two overs as the Black Caps' decision to bowl on a green Basin Reserve wicket after winning the toss paid immediate dividends.
Ashwin is receiving treatment for a left-sided abdominal strain, while Sharma jarred his back while fielding in Adelaide.
The only successful fourth innings Test run chase of more than 300 at Adelaide came in 1902, when the hosts made 315 to beat England.
With five wickets in the shed, India will feel hugely confident of setting Australia an unreachable target.
New Zealand beat Pakistan by 123 runs in Abu Dhabi on Friday to clinch a 2-1 series victory, their first away to Pakistan since 1969.
Cricket Australia had originally pencilled in the ongoing opening Test of the current series as a pink-ball match, but India baulked at the idea.
Shimron Hetmyer top-scored with 39 as the West Indies were bowled out in less than an hour's play after beginning the third day at 75-5.
Paine has also pledged to shake opponents' hands before each match as a mark of respect in a bid to change the culture within the side.
The big total from the home batsmen was a reality check for India's bowlers, who failed to impress.
England only managed to add 24 runs with Moeen Ali falling for 33, Stuart Broad for a duck and Jack Leach 2.
West Indies stuttered to 187-6 by tea on the second day after Hetmyer hit 63 off 47 balls in a quickfire counter-attack at the Zahur Ahmed Chowdhury Stadium.
New Zealand pulled off an extraordinary four-run win in Abu Dhabi on Monday as they defended a modest target of 176.
New Zealand bowlers led by Patel (5-59), fast bowler Neil Wagner (2-27) and Ish Sodhi (2-37) bowled out Pakistan for 171 on a drama-packed fourth day at Sheikh Zayed Stadium.
Pakistan, set 176 to win the match, were 130 for four at the break with Azhar Ali unbeaten on 40 after an innings-reviving stand of 82 with Shafiq in the first session.