The West End Redbacks ended Day Three at 1-49, in pursuit of 331 for the win they need to stay in the hunt for the Shield final.
A Tasmanian batting collapse revived SA's ambitions after the Redbacks had conceded a 166-run deficit on the first innings.
Tasmania further extended that margin to 249 only to lose six top-order wickets for 17 runs.
Fast bowler Daniel Worrall was the chief destroyer with 4-17, as the Tigers - who began the second-last round in second - were cleaned up for 164.
Nick Winter, Kane Richardson and Joe Mennie all chipped in with the ball with two wickets apiece.
Opener Jake Doran top-scored with 53 for the home side, before his dismissal started a slide which included ducks for key batsmen George Bailey and Matthew Wade.
South Australia lost only opener Conor McInerney during a testing 23-over session to conclude the day's play.
CARNAGE in Tassie! The Tigers lose 4-5 😮 🔥— West End Redbacks (@WestEndRedbacks) March 7, 2018
They're 6/99, with a lead of 265. #SheffieldShield