Kohli made 169 before chasing a wide delivery from Mitchell Johnson and edging behind, where Brad Haddin made a diving catch to nick it away from Shane Watson, and India ended day three on 462-8, still 68 runs behind after resuming on 108-1.
Together with Ajinkya Rahane, who made 147, Kohli had steered India away from danger in a partnership worth 262.
However, once that was broken, Australia were able to make inroads as Nathan Lyon and Ryan Harris picked up late wickets.
Dimuth Karunaratne’s maiden Test hundred helped Sri Lanka fight back against New Zealand, but their position still looked grim as they followed on in Christchurch.
The tourists, bowled out for 138 on Saturday in reply to New Zealand’s 441, made a much better go of their second innings as Karunaratne – out for a duck in the first innings – made 152 to help them to 293-5 at the close of day three, still 10 runs behind. His knock was the highest score by a Sri Lankan while following on.
New Zealand remained confident of securing a positive result in this first of two Tests, however, after late wickets from Trent Boult – including that of Karunaratne – swung the momentum back the way of the Black Caps.
Sri Lanka resumed on 84 without loss, with Karunaratne on 49 but they lost Kaushal Silva before he could add to his overnight 33. Kumar Sangakkara quickly followed, making just one before he edged behind, this time off Boult.
However, Sri Lanka rebuilt thanks to a watchful innings from Lahiru Thirimanne, who made 25 off 124 balls.
West Indies batsmen Kraigg Braithwaite and Marlon Samuels provided spirited resistance to frustrate South Africa on day three of the second Test at Port Elizabeth.
The pair shared an unbroken partnership of 92 for the third wicket as West Indies reached 147-2 at the close of play in reply to South Africa’s 417-8 declared
Braithwaite (65 not out) and Samuels (60 not out) were both in sparkling form and West Indies will resume 270 runs in arrears.