The 20-year-old needed 2hrs 47mins to get past a player 15 years her senior 7-5, 5-7, 6-3 ahead of the second rubber between world number nine Ashleigh Barty and two-time Grand Slam winner Victoria Azarenka.
"It was an unbelievable match. She played so quick I couldn't touch the ball at times," said Sabalenka. "But when you're playing for your country, you have to fight to the end."
While Belarus are bidding to reach the final for the second time in three attempts, after being beaten by the United States in 2017, Australia are targeting their first decider in 26 years.
It's been an even longer wait between titles, with the last of Australia's seven victories coming in 1974 when Evonne Goolagong led them to a 2-1 win over the Americans.
France or Romania await this weekend's winner.
Stosur, who was selected to open the tie ahead of the higher-ranked Daria Gavrilova, has traditionally struggled to play her best tennis on home soil.
But in an erratic first set, which saw seven service breaks, she broke first to go 2-1 in front.
The big-hitting Sabalenka immediately broke back but serving problems cost her again when she sent down two consecutive double faults to once more hand Stosur the advantage.
But the Australian couldn't capitalise as nerves took their toll.
She regrouped to break and go 5-4 in front on yet another Sabalenka double fault, but the rollercoaster continued until the Belarusian finally held for 6-5 then took the set on her fifth set point.
Sabalenka won the first two games of the second set and looked comfortable, but a gutsy Stosur reeled off three in a row as the crowd roared her on before wrapping up another see-sawing set in which both players again struggled to hold serve.
But as Stosur tired, the Belarusian grabbed the upper hand in the deciding set with a searing crosscourt forehand for a 3-1 lead, and there was no way back for the 35-year-old who failed to add to her record 29 singles victories for her country.
The reverse singles and a doubles are on Sunday.
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