Teenager shines as Britain seize control of BJK Cup tie with Australia
Gibson beat three top-20 players at Indian Wells en route to the quarter-finals last month and was ranked 219 places higher than the teenager at 56.
But Stojsavljevic was in the zone on the hardcourts at Melbourne's John Cain Arena, banging down 12 aces to win 7-6 (7/4), 7-5.
Her teammate Harriet Dart then battled past Kimberly Birrell 4-6, 6-3, 6-3 to give last year's semi-finalists Britain a 2-0 lead.
A doubles and two reverse singles, if required, follow on Saturday.
"It feels amazing. I can't believe it, to be honest," said Stojsavljevic, the 2024 US Open junior champion.
"I can't even remember the last point. I just played. I had amazing support on the sidelines and managed to get through."
Both teams were without key players with seven-time champions Australia missing injured world number 30 Maya Joint.
None of Britain's top four made the long trip with Emma Raducanu, Katie Boulter and Fran Jones opting to focus on the European clay-court season and Sonay Kartal hurt.
It meant Stojsavljevic was thrown into the fray.
The odds were stacked against her, but she broke early to take a 3-1 lead before Gibson rallied to force a tiebreak.
The power-serving Stojsavljevic charged 4-0 ahead and held her nerve to take the opening set.
Gibson slumped 4-1 behind in the second and when the Briton served for the match at 5-3 it looked to be over.
But the Australian fought to stay in the hunt, only to be broken again at 5-6 and there was no way back.
Dart and Birrell, ranked 173 and 80 respectively, shared the first two sets with the decisive game in the third coming at 3-3 when Dart broke before holding serve and breaking again.
"Kim is such an amazing player and such a fighter and I knew it was always going to be really, really tough, especially here in Australia," said Dart.
Seven ties are being played over Friday and Saturday with defending champions Italy facing Japan at home and last year's runners-up the United States travelling to Belgium.
The winners progress to the eight-team finals in China this September.