Betts (59 from 49 balls) starred with the bat for South Australia posting a maiden WNCL half-century, while Schutt collected 4/33 with the ball.
Schutt won the toss earlier in the day at Karen Rolton Oval and elected to bat first, with Tahlia McGrath (19) and Bridget Patterson (34) starting the side off well.
Suzie Bates (23), Tegan McPharlin (32) and Annie O'Neil (21) all made contributions, however the Scorpions looked to be in some trouble at 5/141 in the 37th over when Betts came to the crease.
In an entertaining knock however, Betts struck seven fours and a six in a 93-run partnership alongside Alex Price (25 not out), who also collected three boundaries.
Chasing 243, Tasmania's Meg Phillips anchored the innings with 64 runs from 110 balls, while fellow opener Hollie Armitage compiled 25 before 16-year-old debutante Darcie Brown (1/32) struck.
Two wickets in two balls from Schutt removed Tuesday's hero, Nicola Carey (7) and Emma Thompson-Flint (duck), however Player of the Match Corinne Hall (68 from 57 balls), Sasha Moloney (30) and Maisy Gibson (29) continued the flow of runs.
McGrath (1/47) finally removed Phillips in the 43rd over, thanks to a diving outfield catch by Patterson, putting Tasmania at 5/195 and reinvigorating the home side.
Despite another wicket from Schutt in the 49th over however, the Tigers chased down 243 with six balls to spare.
The Scorpions will head to Perth next to take on Western Australia on Tuesday 21 and Thursday 23 January.