Bumrah claimed career-best figures of 6/33 in the Boxing Day Test to accumulate 45 Test wickets in 2018.
Pant and Sharma scored freely in the final half hour of play and Kohli ultimately decided against forcing Australia to bowl for a third-consecutive day, instead opting to try and take a vital wicket of two. The visitors were struggling at 54/5 in 27 overs in their second essay but had the upper-hand.
India's captain Virat Kohli has been a popular target for Australian fans during this tour.
"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. But he surpassed B Chandresekhar's twin hauls of 6-52 in each innings at this ground in 1977.
Australia's first innings spanned just 66.5 overs. India leads by 346, which already would be the biggest target set in this series so far.
There were, of course, several reasons India did not want to enforce that follow-on.
President Trump makes surprise Christmas visit to troops in Iraq
Trump had planned to spend Christmas at his private club in Florida, but stayed behind in Washington due to the shutdown. The president has been criticized throughout his presidency for failing to travel to troops in the Middle East.
India were reduced to 32-4 with a double break against Kohli and Rahane.
Earlier, Australia sunk to 145-7 at tea.
Travis Head (20) received similar treatment from the white-hot Bumrah just after lunch, with a cracking in-swinger dipping through an enormous gap the left-hander had left between bat and pad, sending bails flying. Pat Cummins (17) and Paine showed some of the resistance that has been typical of Australia's tail of late, but failed to make hugely meaningful totals.
A draw would represent a moral victory for Paine's team, given India have dominated the contest from the moment Kohli won the toss. He was not out 28 with Rishabh Pant on six.
Australia finished 0-8 in response.
Another session that went in India's favour. The next one too turned from the rough, catching Khawaja's inside edge and giving Mayank, at short leg, his second catch of the morning. Both were caught at leg gully. Australia are reeling at 145 for 7 and are still 298 runs behind. They used good footwork against Jadeja to smother the possibility of turn from the rough.
Shaun Marsh showed a lot of discipline and nearly saw through the session but was flummoxed by a superb slower one by Bumrah off the last ball before lunch to be dismissed LBW for 19, leaving Australia in despair. The Australian spectators for the first two days of the Test match constantly chanted racist comments against the Indians.