Kohli eventually went when he was adjudged to have been caught by Peter Handscomb off Pat Cummins in a decision which was given out by the on-field umpires and not changed by the third umpire, with television replays prompting doubts over whether the ball was cleanly gathered.
That began a lower-order collapse as India lost their last five wickets for 35 runs, their tail not taking long to wilt.
On a menacing Perth pitch, Harris had been struck on the helmet by a Jasprit Bumrah bouncer, while Finch retired hurt after being hit by a Mohammed Shami delivery, but is expected to bat again on Monday after no serious damage was discovered.
Harris was then bowled by Bumrah for 20 as the home side also lost Shaun Marsh, Handscomb and Travis Head cheaply before the end of the day.