Australia hoped to avenge their World Cup semi-final loss to the Lionesses in 2023, but European champions England were on top form in the second game of their 'homecoming tour' at sold-out Pride Park.
England bounced back from their defeat to Brazil three days earlier, taking advantage of Alanna Kennedy's 19th-minute red card.
Sarina Wiegman's side have been accused of starting slowly recently, but first half goals from Aggie Beever-Jones and Bronze, on her 34th birthday, ensured they made a fast opening this time.
A celebratory evening for the hosts was marred by the sight of England striker Michelle Agyemang being stretchered off after she went down clutching her knee.
Georgia Stanway put the seal on England's one-sided victory with a stoppage-time penalty.
Australia have just two wins from their first five games under coach Joe Montemurro.