Kyah Simon's 80th-minute finish settled the contest and returned the Matildas to their third straight Cup quarter-final.
But it was Elise Kellond-Knight who picked up her second straight player-of-the-match award for her defensive shielding role in midfield.
Kellond-Knight said she was "super proud" of her team-mates for winning through to the last eight.
"There's a sense of satisfaction, we've worked so hard for this," she said.
At her second World Cup, the accolades aren't new for the 24-year-old.
Kellond-Knight was voted as the tournament's best left-back in 2011 but has been re-cast as a midfielder in 2015.
Stajcic said the award was just reward for "KK" who fought her way into the Matildas first 11 on the eve of the tournament.
"To be honest, she was struggling for a starting spot two or three weeks before the tournament," he said.
"There's a huge internal rivalry we've developed within the team, competition for spots.
"She had to reflect and come back and work harder and we're seeing the results of that now.
"She's a world-class footballer and I think every game she's getting better and better."
That competition for places relegated Simon to the bench against Brazil.
After replacing Michelle Heyman on the hour mark, the Sydney FC striker made sure her presence was felt with a rebounding volley after Lisa De Vanna's shot was spilled.
Stajcic said he was one of the last to react to the goal.
"I didn't actually jump until I saw the net rattle, I just wanted to make sure," he said.
"She certainly had to work hard to get there, it wasn't an easy tap in.
"A magic moment like that gets us through to the quarter final ... that's what makes great players and Kyah Simon is a great player."
Kellond-Knight beamed at the realisation she was part of the first senior Australian side to win a knockout match at a World Cup.
"The pride that comes through knowing that is tremendous," she said.
"I'm super satisfied with the performance we put out today to come away with the result."
