Young bowlers Nick Winter and Wes Agar swung momentum back in favour of the Redbacks after the Tigers looked likely to chase down the set total of 337 thanks to a second consecutive century in the game from Alex Doolan.
Doolan provided resistance in both innings but then feathered an edge to the wicketkeeper from the bowling of Wes Agar (3-49) which turned the tide, with McDermott out soon after.
Tasmanian stalwart George Bailey (32 not out) and tailender Lawrence Neil-Smith (one not out) navigated the last seven overs to deny SA.
The Redbacks' victory prospects appeared remote until Doolan, Ben McDermott (42), Test captain Tim Paine (one), Jackson Bird (six) and Alex Pyecroft (13) fell in rapid succession.
Doolan, who faced 604 balls across both innings, had earlier put on a 95-run stand with Beau Webster (40) as the Tasmania's sought a second win of the Shield season.
Test batsman Matthew Wade cracked 38 from 37 balls before being caught attempting a reverse sweep, before Doolan and McDermott featured in a 53-run partnership.
Tasmania required 74 runs with five wickets in hand from the final 15 overs in the last hour, with stalwarts Bailey and Paine at the crease.
But the Test skipper was soon bowled by Nick Winter (2-41) and Bird was caught at deep square leg four overs later, leaving SA seeking three wickets in the final 12 overs to win.
Bailey found an ally in Pyecroft, whose cameo again lifted Tasmanian hopes.
But Pyecroft was dismissed with 41 balls remaining, forcing Bailey and No.10 Neil-Smith to shut up shop and deny SA a deserved win.
The match was riddled with sporting declarations after South Australia’s record opening partnership between Jake Weatherald (198) and Henry Hunt (132) helped them to 6d-409 in their first innings.
After Tasmania declared behind, the Redbacks then came out for only four overs on the morning of the final day to set up the 337-run chase for the visiting team.
South Australia will next play Western Australia in a Marsh One-Day Cup match at Karen Rolton Oval on Sunday, before a Sheffield Shield match against the same opposition at the WACA on Friday, 29 November.