It was a morale boosting win for Australia, who have endured a dismal run of late in limited overs cricket.
The second match of the three-match series will be played in Melbourne on Friday.
Dhawan got India off to a quick start, putting on 35 off 25 balls for the opening stand with Rohit Sharma (7).
The latter was caught at long on off Jason Behrendorff (1-43) in a bid to accelerate his strike-rate.
In keeping with the strategy used during the T20I series in England, KL Rahul (13) came out to bat at number three.
Dhawan and Rahul put on 46 runs for the second wicket, but it was mostly down to the left-hander’s belligerence. He hit ten fours and two sixes overall, and reached his ninth T20I half-century off only 28 balls.
Rahul though was patchy at best and struggled for timing. He was stumped off Adam Zampa (2-22) in the ninth over, with the leg spinner also accounting for skipper Virat Kohli (4) who never really got going coming down at number four.
Zampa should have had a third wicket but he dropped a return chance from Dhawan (on 65). The batsman enjoyed another life at 74, when substitute Nathan Coulter-Nile put him down at square leg off Billy Stanlake (1-27).
The asking rate was climbing up and it took a toll on Dhawan, who finally holed out of Stanlake, leaving Rishabh Pant (20 off 15 balls) and Karthik with a mountain to climb.
They nearly achieved the impossible, putting on 51 off a mere 24 balls, toying around with the Australian bowling. But what is becoming increasingly frustrating with Pant, he played yet another loose and unnecessary stroke, throwing his wicket away.
It left Karthik to finish off things, but he found the going tough without enough support from the other end. With 13 needed off 6 balls, Krunal Pandya (2) and Karthik holed out off consecutive deliveries off Marcus Stoinis (2-27).
This was after Maxwell hit four sixes in a whirlwind knock before rain came, after Chris Lynn scored 37 runs off 20 balls to help Australia recover from a slow start.
India had won the toss and opted to bowl on a surprisingly green wicket.
Kuldeep Yadav was picked ahead of Yuzvendra Chahal as expected, while the hosts also included a spinner in Adam Zampa and left out medium pacer Coulter-Nile.
Bhuvneshwar Kumar (0-15) and Jasprit Bumrah (1-21) got India off to a good start but Aaron Finch (27) pulled things back a bit.
Khaleel Ahmed (1-42) got the initial breakthough as D’Arcy Short (7) was caught at long on. The left-arm pacer proved expensive thereafter.
Kuldeep took a sensational catch running back to dismiss Short, and from there onwards, he had a marked influence on the game.
Later he also picked up 2-24 in his four-over spell. He dismissed Finch and then sent back Lynn as well, thus reducing Australia to 75-3 in the 11th over.
Apart from Yadav’s stunning effort, India were rather poor in the field. Skipper Kohli had dropped Finch in the fourth over, a straight chance spilled at cover, and then miss fielded later on as well.
Ahmed too dropped Stoinis (33 not out off 19 balls) late in the Australian innings, while Maxwell should have been run-out.
Finch took advantage of the reprieve and put on 40 runs off 26 balls with Lynn for the second wicket.
But it was Maxwell who stole the show with his belligerent hitting as Australia crossed 150 in the 16th over. He put on 78 runs off 37 balls with Stoinis as Indian bowlers proved expensive even before the death overs began.
Krunal Pandya had a forgettable day with both bat and ball. He went for 0-55 in his four overs before rain intervened, sucking out momentum from the Australian innings and leaving them only five deliveries on resumption of play.
Bumrah bowled cleverly, conceding only five runs after the disruption, while Australia still finished with a challenging total on the board.