The Azzurre, back at the World Cup after a 20-year absence, prevailed when striker Bonansea headed home from a free kick five minutes into stoppage time.
Australia captain Sam Kerr had opened the scoring by firing home on the rebound after her penalty was saved by Italy keeper Laura Giuliani in the 22nd minute.
Bonansea, however, equalised 11 minutes into the second half after a defensive blunder on the edge of the box.
"I’m very happy with this result. But it only holds real value if we get out of the group, so let’s wait and see how great it is," Italy coach Milena Bertolini said.
"In the first half, we didn’t showcase our qualities. Australia were strong but we didn’t do ourselves justice. In the second half, we wanted to show that we were superior to Australia and we improved a lot of things about our style of play.”
Australia coach Ante Milicic vowed to stay true to the Matildas' style despite the defeat.
"We are not going to change our style because we got caught at the back (for Italy's first goal)," he said.
"We will always build up from the back."
Jamaica play Brazil in the other Group C game on Sunday.
(Reporting by Julien Pretot; Editing by Toby Davis)
