His likely replacement, rookie Will Pucovski, was ruled out with concussion after being struck on the helmet by a bouncer in a warm-up match last week.
Kohli will only lead the side in the first of the four Tests -- starting in Adelaide on December 17 -- before returning home for the birth of his first child.
Once the Indian skipper was out 11 balls from the finish, his team's hopes diminished and they failed to sweep the series after wins in the opening two games.
Unchanged India are chasing a 3-0 series sweep in Australia for the second time in five years after losing the preceding one-day international series 2-1.
India needed 25 runs off the last two overs and Pandya delivered in style, smashing two sixes in the final over to secure a six-wicket victory with two balls to spare.
Starc was withdrawn on compassionate grounds due to a family illness and will miss the final two matches of the series, while Hazlewood has a back injury.