Chasing 185 for victory, Australia were in big trouble at 5-57 entering the fourth and final day.
A 44-run partnership between Ellyse Perry (31) and Sarah Elliott (29) breathed life into the Southern Stars' run chase as they moved to 5-99.
But Perry's dismissal sparked a collapse of 3-7 as Australia were bowled out for 123 shortly before lunch.
The Test win gave England six points in the multi-format series, which also features three one-dayers and three Twenty20s.
With each one-dayer and T20 match only worth two points, Australia need to win five of the remaining six matches to regain the urn.
That will be a tough task against an England side now brimming with confidence following their early success on enemy territory.
But the visitors will need to find an answer to star Australian allrounder Perry, who was the clear standout during the one-off Test.
Perry took 3-41 and 5-38 with the ball, and scored 71 and 31 with the bat to give the Stars a sniff of victory.
England speedster Anya Shrubsole was the hero on the final day, snaring three wickets, including the key scalp of Perry, to put the Stars to the sword.
Kate Cross took three wickets late on Sunday to put England on track for victory, with Katherine Brunt wrapping up the triumph when she clean bowled Elliott.