Australia staged a remarkable three-goal comeback to beat Brazil 3-2 and pick up three crucial points in Group C at the Women's World Cup in Montpellier on Thursday.

Brazil raced out to a 2-0 lead in the first half, with Marta converting a penalty in the 27th minute and Cristiane beating her defender to a cross just over 10 minutes later to double her team's lead with an excellent headed finish.

The goal from Marta, a fitness doubt coming into the match, extended her record as the all-time Women's World Cup goals leader to 16 and made the 33-year-old the first women to score at five different editions of the tournament.

Australia struck back just before half-time, though, when Caitlin Foord poked home at the back post from a flicked on header to halve the deficit and send the Matildas into the dressing room trailing 2-1.

Shortly before the hour mark, Australia were back on level terms. A Chloe Logarzo shot from distance crossed up Brazil goalkeeper Barbara and found its way into the back of the net to make it 2-2.

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Australia then took the lead through a Brazil own goal thanks to a favorable decision from VAR, as an initial offside call against Sam Kerr was overturned following the referee's consultation with the pitchside monitor to complete the Matildas' three-goal rally.

"Tonight is one of the finest Australian performances I've seen. The players deserve all the rewards they were given tonight. I'm really delighted for them," said Australia coach Ante Milicic.

"They never stopped believing, and the coaching staff never stopped believing. We kept on believing in our style and in our football and it's an Australian performance to be proud of."

Brazil pushed hard for an equaliser, but the Australian defence stood firm to see out the result and join Brazil and Italy, who have yet to play their second match, on three points in the group.

Marta, who had already scored at the 2003, 2007, 2011 and 2015 World Cups, admitted her record goal was bittersweet given the result.

"It would feel much better with a win," Marta said. "It's one more detail written in the history of women's football. I am honoured, but there's more to do at this tournament."

The result marks just the second time in Women's World Cup history that a team came back from two goals down to win a match and gives Australia a very solid chance of reaching the round of 16.