Australia went down in the dying minutes of injury time as Italian midfielder Bonansea headed home her second goal of the game to give Italy their dream start to the tournament, and the Matildas plenty to think about.
The Italians grabbed the equalizer in the 56th minute when a mistake on Australia's backline delivered Bonansea with clear space down the left wing.
And Australia were again the makers of their own undoing deep into injury-time as goalkeeper Lydia Williams was caught all at sea from a deep corner delivery to allow Bonansea to head home the victor.
The Aussies dominated most of the play controlling 64 per cent of the possession and getting away 16 shots to Italy's 6. Chances for Kerr, mainly created by wingbacks Ellie Carpenter and Steph Catley, were few and far behind, and the Australians couldn't make their opportunities matter.
In another fixture of Group C, 2007 finalists Brazil enjoyed a comprehensive 3-0 victory over Jamaica, as Cristiane completed a hat trick.
"We have the potential to get through, we've just chose to go the long way, the hard way about it".
Handed a chance to break her World Cup duck, Kerr's initial effort was saved by Laura Giuliani, but she tapped in the rebound to spark Australia's attack.
The score stayed level for the majority of the second half before Italy threatened with a free kick in stoppage time. In Group D, Nikita Parris and Ellen White scored for England in their 2-1 win over rivals Scotland, who had their consolation through Claire Emslie. But that was as close as they would come to stealing a point on Sunday.