Chasing 187 to sweep the three-match series, Kohli kept India in the hunt before falling in the penultimate over in the first full-capacity cricket match in Australia since the COVID-19 outbreak.
Steve Smith dropped him when Kohli was on nine, and Andrew Tye spilled a return catch when the batsman, who also survived a stumping opportunity later on, was on 19.
Leg-spinner Mitchell Swepson (3-23), who dismissed Sanju Samson and Shreyas Iyer in his final over to hurt India's chase, was adjudged man-of-the-match.
Earlier, Matthew Wade smashed a career-best 80 as he and Glenn Maxwell powered Australia to 186-5.
Maxwell joined Wade the middle and started messing with the bowlers' line and length by repeatedly playing the trademark reverse shots.
The sloppy tourists did not help their cause either.
Maxwell was caught at 18 at midwicket but was recalled after a replay confirmed spinner Yuzvendra Chahal had overstepped.
T Natarajan eventually bowled Maxwell for 54 off 36 balls.
Australia had won the preceding ODI series 2-1.