Needing to manufacture the highest innings of a bowler-dominated match, the reigning champions recovered from 5-88 to chase down the target of 217.
Cult hero Ben Manenti was one of the heroes, scoring an unbeaten 49 from No.8.
Manenti (3-26) also took the crucial wickets of Beau Webster, Tim Ward and Brad Hope in Tasmania's second innings to help bowl the Tigers out for 184.
He put on a crucial 71-run stand with Liam Scott (53) to steer the visitors out of trouble
SA were winless from their first three games of the season, losing two and drawing the other.
The result came against stiff opposition, with Australia's incumbent Test No.6 Beau Webster starring with the ball to ensure he remains in contention to keep his spot.
After claiming 5-50 in the first innings, Webster backed it up with 3-73 in the losing cause.
Two of his victims were Travis Head (15) and Alex Carey (23), his Australian teammates.
Head edged Webster to slips from a ripping off-cutter, continuing his disappointing run of form leading into the Ashes.
Since smashing a blazing ODI century against South Africa in August, his highest score has been 31 in 11 innings.
Although Webster got the better of his Test teammates, he was taken apart by Manenti and Scott, with his wickets coming at more than six runs an over.
Manenti was thrilled with the win.
"We've been pretty successful down here the last couple of years. It's a place we love to come and play at," he said.
"Probably rode the game a bit, it was a tricky wicket early.
"We needed it. We've been close the last couple of weeks, playing some really good cricket, we've just lost patches."
SA will return to Adelaide Oval for their next match against Western Australia, starting on November 22.
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