Rishabh Pant helped himself to 18 runs off the first over after lunch, delivered by Lyon, but went down swinging to the next delivery he faced from the tweaker. At the other end, vice-captain Rahane was also solid, surviving a review for a bat-pad catch off Lyon.
Australia captain Tim Paine pointed out that Cheteshwar Pujara was the difference between the two sides in Adelaide. It was his 13th five-wicket haul in Tests.
On India's sixth win in Australia, Kohli said, "These things happen in Tests, ups and downs through a game".
The highest successful run chase in a Test at the venue is Australia's 6-315, achieved in 1902 against England, while no side has chased down more than 236 since West Indies in 1982.
Australia needs 219 more runs to claim an unlikely victory, which is already looking like a step too far on a fifth-day pitch, which will have plenty of turn for Ashwin to exploit.
A sliver of hope will come from Western Australia in a domestic match last month hitting 313 to beat South Australia here with Shaun Marsh - who is playing in the Test - scoring 163 not out. They fought really well but we executed our plans well. "We definitely still believe we can win this".
Australia resumed after the break 137 runs shy of the victory target, but reduced the margin by only one run before Paine flashed his bat at a short ball to be caught by fellow wicketkeeper Rishabh Pant after skying a top edge.
They had reached 291 when last man Josh Hazlewood was caught off spinner Ravichandran Ashwin.
Marsh edged a beauty from Jasprit Bumrah (3-67) behind before Paine miscued a pull shot off the same bowler and Pant then notched his record-equaling catch when Starc looked to attack Mohammed Shami (3-65). Australia desperately needed the experienced Usman Khawaja to stick around but he chose to run down the pitch to an Ashwin delivery on eight and sliced it to Rohit Sharma, who took a hard catch.
Kohli did not deny feeling the pressure on the last day of the match.
"I thought the bowlers were brilliant in the first innings, defending 250, the discipline was magnificent". The aggressive Pant went in search of quick runs as India's lead approached 300, whacking three fours and a six in succession off one Lyon over.
It was his 10th Test half-century overall.