After Australia's middle order collapsed in their second innings, England's opening pair began the chase with a solid 55-run partnership.
The start gave the home fans hope of victory until spinner Ashleigh Gardner took three wickets.
Tammy Beaumont, who had smashed 208 in the first innings, was the first to fall, dismissed off the first ball of Gardner's spell when she nicked to first slip.
"It was a lucky ball to get Tammy, she made 200 yesterday so I'll take that," Gardner told Sky Sports.
"Bowling in partnerships was just as important to put pressure back on England because I'm sure they came out quite confident after bowling us out."
Medium pacer Tahlia McGrath got in on the act to dismiss opener Emma Lamb while Nat Sciver-Brunt was caught off Gardner as England's top three fell within a span of 11 runs.
Skipper Heather Knight became the second English batter after Lamb to be dismissed lbw, with both failing to get the Decision Review System (DRS) to overturn the umpire's decision.
Danni Wyatt and Sophia Dunkley then steadied the ship with a patient 37-run partnership. But Kim Garth picked up her first test wicket when she dismissed Dunkley, caught behind off an outswinger.
Wyatt and Kate Cross were at the crease at stumps with England needing 152 more runs to win.
ECCLESTONE TAKES FIVE
Earlier, Australia had resumed on 82-0 after their openers survived 19 overs on Saturday but Phoebe Litchfield (46) was the first to fall, misreading an inswinger from Cross and leaving the delivery which nipped back in and hit her off stump.
All-rounder Ellyse Perry (25) did not last long either when she played on a delivery from Lauren Filer.
Spinner Sophie Ecclestone dismissed the dangerous Beth Mooney, another player who played on to get bowled for 86, while skipper Alyssa Healy contributed with a fifty but fell when she chipped a full toss to midwicket for a catch.
Cross and Filer picked up two wickets each before Ecclestone took the final wicket just before the new ball was due for her second five-wicket haul in the match as Australia were dismissed for 257.
"We probably didn't bat to our full potential in our second innings but you have seen in this final innings how hard it is to bat. Some are spinning and some are rolling," Gardner added.
"We spoke at tea how crucial those runs were from Alyssa Healy and it was a monkey off her back. She led from the front and took it into her keeping as well."