The match ended in the hosts' favour by 49 runs and gave them a 1-0 lead in the four-match series.
Kohli was the stand-in skipper for the first Test, his first-time leading the Indian team in the longer format, even as Mahendra Singh Dhoni is expected to lead the side in the second Test in Brisbane starting December 17.
In becoming only the fourth Indian to score twin hundreds in a Test, the star batsman led from the front as he almost took the visitors to an improbable victory.
"He knows you have to stand up for yourself and be prepared to compete in the middle. He has done that not just with his captaincy but also with his bat. I liked the fact that they went for the debutant Karn Sharma in the XI thinking that leg-spinners do well here.
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"I liked the intent even though I think R Ashwin would have bowled better than he did. I think this will be a really good Test series. We have almost been waiting for a contest like this.
Australia returned to top-level cricket with this match after the tragic death of Phillip Hughes. The first three days of the game saw some intense emotional moments.
But as the match slowly progressed towards its climax and tensions rose, there were a couple altercations between the players of two sides.
The aggression displayed on the field was perhaps a sign that normalcy has returned to Australian cricket.