Rahul soon completed his half-century in the 28th over and India reached 106-3 and were in control.
Kohli was eventually caught at midwicket off Hazlewood and received a standing ovation.
Rahul hit the winning six with Hardik Pandya standing at the other end.
“I had just showered and was expecting to put my feet up for half an hour after fielding, but I had to come in,” Rahul said of arriving at the crease unexpectedly early.
“Virat said that I should play like Test cricket for some time.”
Australia captain Pat Cummins admitted that his team “fell 50 runs behind”.
“The spinners were tough to face and it was the kind of wicket you had to adapt to.”
Five-time champions Australia won the toss and decided to bat first but faltered.
Jadeja was excellent in his 10 overs in left arm spin bowling. Fellow spinner Kuldeep Yadav and fast bowler Jasprit Bumrah took two wickets each.
Australia lost an early wicket before veteran batsmen David Warner (41) and Steve Smith (46) stitched a 69-run partnership.
