At the breather, Australia were trailing India by 298 runs with their skipper Tim Paine unbeaten on 22 and pacer Mitchell Starc (5 not out) giving him company.
After the impressive Jasprit Bumrah (6-33) exposed Australia's brittle batting to dismiss the hosts for just 151, Kohli resisted the temptation to ask Tim Paine's men to bat again.
At stumps, the visitors had a formidable 346 cushion with five wickets left and two days to play as they inch closer to a first ever series win Down Under. This is also his career best performance in Tests, bettering the 5/54 which he claimed against South Africa at Johannesburg in January.
His scalps also included opener Hanuma Vihari (13) and vice-captain Ajinkya Rahane (1).
He picked 6/33 in Australia's first innings.
Cummins was Australia's best bowler in India's first innings 443-7 declared and was chief destroyer on Friday, claiming an incredible 4-3 off his first four overs.
Rishabh Pant and Mayank Agarwal had a clear plan of getting to a lead of 400 as quickly as possible when they walked out for the start of the day's play and both played their part in pushing the game forward, with Agarwal taking a liking to the spin of Nathan Lyon, hitting two consecutive sixes in one over.
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The Test match then took a huge turn after Indian skipper Virat Kohli decided not to enforce the follow-on.
"You dare to dream", said Cummins on Australia's chances of avoiding defeat, while defending what was arguably one of their worst battings displays in recent times.
India took nearly two days to score their first innings total of 443, but needed less than two sessions to rip through Australia's top-order batting.
Pace bowler Ishant Sharma removed Finch caught at short leg. Harris fell three overs later, caught at fine leg off Bumrah's bowling after the batsman top-edged a pull shot.
India declared early on day four of the third test in the four-match series, giving Australia a sniff at victory.
Shaun Marsh was then deceived on the last ball of the morning by some late swing from Bumrah to be plum lbw. Travis Head again looked good while making 20, but messed it up by playing all over a reverse-swinging delivery from Bumrah and having his stumps shattered.