David Warner started the first innings with two back to back boundaries off Bhuvneshwar Kumar but failed to build the innings as he was sent back by his SRH teammate for eight runs.
Opening batsman Shikhar Dhawan, who missed the one-day worldwide series against the same opponents to be with his ailing wife, remained unbeaten on 15 off 12 deliveries.
After a delay of nearly two hours, India were given six overs to chase 48 which they hunted down with three balls to spare.
Yuzvendra Chahal (1-23) provided India with the breakthrough when he once again removed Glenn Maxwell (17) but Finch kept Australia in the game before he was accounted by Kuldeep Yadav (2-16). Then rain stopped the match.
India won the one-day series against Australia 4-1 to reclaim the No.1 spot in the ICC rankings and the hosts would be aiming for a flawless finish to the campaign with the T20I series.
"Smith landed awkwardly on his right shoulder while diving in the field during the fifth ODI against India last Sunday", team doctor Richard Saw stated in a press release by Cricket Australia.
The top scorer in the innings was Aaron Finch, who opening the batting, hit 42 off 30 balls. The match was being played at Ranchi, which is MS Dhoni's home ground. However, given his record in this format and the fact that his last T20I came 18 months back against India at the World T20, this could be a blessing in disguise for the visitors.
The squad has five new faces in Moises Henriques, Tim Paine, Jason Behrendoff, Andrew Tye and Daniel Christian. Jasprit Bumrah meanwhile had figures of 2 for 17 in the 3 overs he bowled.
His return from a shoulder injury has not been up to the mark as he did not get a chance in the ODI series but Rahul has been retained for the T20I series and he would look to make it count. For India, Axar Patel and Ajinkya Rahane sat out and they were replaced by Kuldeep Yadav and Shikhar Dhawan.