The tourists set a victory target of 305 after knocks of 85 and 72 from George Bailey and Aaron Finch respectively, as well as late flurries from James Faulkner (27 off 22 balls) and Clint McKay (11* off 4).
Faulkner took the first of his three wickets in the seventh over when a full delivery was nicked through to Brad Haddin by Shikhar Dhawan (7).
He was so pumped up by the dismissal that he went for a high-five with Haddin but missed, poking his wicket-keeper in the eye.
Kohli's dogged innings of 61 and any Indian hope of a late surge towards 300 came to an end in the 35th over, with the remaining five batsmen failing to score more than 20.
The second match takes place in Jaipur on Wednesday.