Rohit, Kohli fall as India 33-3 chasing 340 against Australia
An unrelenting Pat Cummins took 2-10 off nine overs, removing Rohit for nine and KL Rahul for a duck in quick succession on the fifth morning at the Melbourne Cricket Ground.
India's hopes of reaching their target of 340 then suffered a further blow when Kohli departed off the final ball before the interval.
Kohil had scored just five when he tried to drive a wide delivery from Mitchell Starc and edged to Usman Khawaja at first slip.
Opener Yashasvi Jaiswal was 14 not out at the break after grinding for 83 balls, with India's run rate reduced to a stagnant 1.26 by accurate Australian bowling.
India need a record fourth-innings Melbourne run chase of 340 to go 2-1 up in the series, after Australia posted 234 in their second innings.
More likely they will need to bat out the rest of the day to salvage a draw and keep the series locked at 1-1 going into Sydney's final Test, which begins on Friday.
Australia captain Cummins continued a brilliant all-round Test, after scoring 49 and 41 with that bat, by making a double breakthrough in the 17th over.
Opposing skipper Rohit had batted with discipline but fell when attempting an aggressive whip across the line that flew quickly to Mitchell Marsh at gully.
The out-of-form Rohit has failed to go past 10 in three Tests after missing the opener in Perth for the birth of his second child.
India were 25-1 at that stage and it quickly became 25-2 as Rahul fell five balls later, snicking a rising Cummins delivery to Khawaja at first slip.
Earlier, Australia's stubborn final-wicket stand of 61 was finally broken 10 balls into the day when Nathan Lyon was bowled by Jasprit Bumrah for 41.
Last man Scott Boland scored a dogged not out on 15 off 74 balls.
Marnus Labuschagne top scored for the home side on Sunday with 70, helping to revive an innings that was faltering at 91-6.
The exceptional Bumrah took 5-57 to give the fast bowler nine wickets for the Test and 30 across the first four Tests of the series.