The victory extended the holders' lead to an unassailable 8-4 in the seven-match points-based series comprising one test, three one-day and three T20 internationals.
It meant England can only draw the series 8-8 if they win the final two T20s over the next week. Australia won the last series, which was staged in England, 10-6 in 2015.
England, knowing they had to triumph to stay in the hunt to regain the Ashes that they had last won in 2013-14 in Australia, made a terrible start, slumping to 16-4 before Dani Wyatt's 36-ball 50 helped them to a respectable total.
Left-hander Mooney, though, played superbly, recording the highest score by an Australian on home soil in women's T20 internationals, hitting 11 fours and two sixes in a dazzling 56-ball knock.
Australia captain Rachael Haynes said afterwards: "It was so nice for the team to come out and play like that. For Beth Mooney to come out and make a statement like that was fantastic."
Her England counterpart Heather Knight conceded: "Credit to Australia. Beth Mooney played an outstanding innings and took the game away. Everything we tried, we couldn't get her out."
(Reporting by Ian Chadband; Editing by Toby Davis)
