Green, the man of the match, smashed eight fours and four sixes after the Aussies lost skipper Aaron Finch for 22 in the third over.
The right-hand all-rounder stitched a 70-run partnership with Steve Smith before he was caught by Virat Kohli off Axar Patel.
Smith, 35 off 25, departed in the next over followed by Glenn Maxwell's wicket in the same over for one.
However, Wade displayed a powerful batting performance, hitting six fours and two sixes helping the visitors to chase down the target in the 19th over.
Tim David, who made his T20I debut, scored 18 off 14 balls while Josh Inglis made 17 off 10 balls.
Pat Cummins hit the winning shot off Yuzvendra Chahal to reach 211-6.
Patel picked three for 17 while Umesh Yadav took 2 wickets for 27.
It was Australia's second highest successful chase in the T20 format and the highest against India.
"At this ground it didn't worry us too much. Anywhere between 12-14 an over at the back end on a wicket like this and an outfield like this we back ourselves to get them," Wade said.
In the first half, the visitors won the toss and opted to field, with all-rounder Hardik Pandya smashing a rapid 71 to help the hosts post a challenging total of 208-6.
Opener KL Rahul also made a half-century while Suryakumar Yadav hit 46 after losing skipper Rohit Sharma for 11 and star batsman Kohli for two in the powerplay.
Rahul compiled 55 off 35 balls, hitting four fours and three sixes before he was caught by Nathan Ellis off Josh Hazlewood at deep square-leg.
Yadav hit four sixes in his 25-ball stay before he was caught behind off Green.
But Pandya increased the scoring rate further, hammering 71 off 30 deliveries.
The 28-year-old all-rounder hit seven fours and five sixes including three back-to-back hits in the last three balls of the innings.
Ellis was the most successful bowler and finished with figures of three for 30.
The three-match series is being used as a warm-up for next month's T20 World Cup in Australia.
The visitors are playing without pace spearhead Mitchell Starc and all-rounders Marcus Stoinis and Mitchell Marsh due to injuries.
India are looking to bounce back after crashing out of the Asia Cup this month before the knockout stage.
The teams will meet on Friday for the second T20I in Nagpur.