Rossouw, making his T20 international debut, smashed three sixes and seven fours in his 50-ball knock to get the Proteas home with 145 for three. The Australians could only manage 144 for six off their 20 overs after winning the toss at the Adelaide Oval.
South Africa lost Reeza Hendricks off the third ball of the innings but Rossouw combined with wicketkeeper Quinton de Kock (46 off 39 balls) to put on a match-winning 129-run partnership for the second wicket.
By the time Rossouw skied Pat Cummins to Ben Cutting in the 15th over, the Proteas only needed 16 runs for victory.
“Fortunately for me, tonight was my night,” Rossouw said.
“You’ve got to be on top of your game and if it’s your night you have to capitalise.
“It’s very nice batting with Quinton, it takes a lot of pressure off your shoulders.
“Every opportunity I get I want to shine and luckily for me tonight was my night.”
The Proteas’ well-drilled bowling and outstanding fielding kept the Australians under pressure with Shane Watson, in his first international appearance for seven months, top-scoring with 47 off 36 balls.
Watson’s innings, with support from James Faulkner (41 not out), revived the home side after a poor start.
The Australians struggled to 67 for four after 10 overs with both their international debutants, Ben Dunk and Nathan Reardon, failing.
Dunk made just two and Reardon four after opening batsmen Aaron Finch (14) and Cameron White (24 from 21 balls) had begun briskly.
“I thought we were pretty disappointing with the bat. It was a beautiful wicket and the guys at the top of the order didn’t really go on and get that big score,” said Finch, Australia’s T20 captain.
“There were a lot of starts and it was disappointing to get out like that, but the way we fought with the ball was pretty impressive.”
The openers took 27 from the initial three overs before tight bowling and brilliant fielding stifled the Aussies.
White cracked an uppish cut only for Farhaan Behardien to snaffle a superb overhead catch at point, while Reardon’s debut dig was spectacularly cut short by Proteas wicketkeeper de Kock.
Paceman Kyle Abbott was the pick of the South African bowlers with three for 21 off four overs.
Australia must win the second match in Melbourne on Friday to keep the series alive until Sunday’s third and final game in Sydney.