Without a win in their opening two matches, Tasmania maintained a healthy run-rate throughout with Ben Hilfenhaus edging the winning runs off the penultimate delivery.
An embarrassing missed run out opportunity by Kane Richardson in the final over cost the South Australians dearly, the quick bizarrely failing to hit the stumps at the non-striker's end when Tasmania's Ben Laughlin backed up too far.
Richardson had plenty of time to knock the bails off with ball in hand but threw the ball at the stumps at close range only to miss.
The duo then closed out the match for the Tigers.
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South Australia accumulated runs at a good rate throughout their innings to finish with 6-268; 32 runs higher than the best score of the season to date set by Victoria.
Opener Michael Klinger struck an unbeaten 140 in a sparkling knock which included three sixes.
Cooper was dismissed for 65 off 71 deliveries, in a fluent innings which included four fours and two sixes.
Tasmania rallied in the second half of the innings, taking wickets at regular intervals to help temper the run rate until South Australia's late surge.
South Australia captain Johan Botha, who was cited for a suspect bowling action on Monday, looked impressive until being dismissed for 16 off 25 balls.
Off-spinner Rose was the best of the Tasmania bowlers collecting 2-33 on debut off his allotted ten overs, having made the move from Victoria during the winter.
Tasmania were marginally off the required run rate for much of the innings.
Laughlin said a return to Tasmania reinvigorated the team after successive defeats.
"We had a chat about batting against spin which worked well for us today, and also about developing partnerships though we are still waiting to get a big one," he said.
"At half-time I thought they were probably 30-40 runs above par based on recent matches here, but in the end I think we proved it was a pretty good wicket for batting."
South Australia centurion Michael Klinger refused to lay any blame on Richardson despite his poor missed run out chance.
"I wouldn't blame Richo on that one," he said.
"They only needed a run a ball at that point. Personally, I don't think they should have got near our score if we bowled to our plan but we didn't quite do that."