After sending down the fourth ball of the over to Dawid Malan, which was jabbed back towards him, Starc turned and pointed at Buttler and the crease.
He appeared to say something to Buttler which was not audible, but the stump mic did pick up Buttler saying, "don't think I was".
Finch said he had not seen the moment occur in the middle but that a first warning was probably the way to go, although he was not broadly in favour of the mode of dismissal.
"I wasn't aware, had no idea about that," he said. "I think if batters get a warning then it's fair game after that. That would go for most teams, I assume, if you give the batter a warning that they are getting a little too much ground before the ball is bowled. But I'm not a big fan of it, to be fair, personally."
Asked if he would be comfortable with a bowler completing a run out, Finch said: "Great question..."
"No, I am calling the batsman back," he said. "No one wants to see them in the game because they always create such a talking point when it should be about the battle between bat and ball and watching great games of cricket. They always seem to happen at unsavoury times."
On that occasion, Buttler later admitted he would probably have taken a different approach in a match of more significance.