Rain and a stubborn stand from Ferguson threatened to deny the Tigers at Blundstone Arena but former Test quick Jackson Bird struck twice in the shadows of stumps to help the hosts to outright points.
Set a mammoth 403 for victory, the Redbacks were eventually bowled out for 213.
Players were forced from the field for almost two hours in Friday's third session with SA teetering at 7/192.
Once they returned, Bird (4/65) had Nick Winter (26) edging before trapping Ferguson in front for 111.
Ferguson, though, was a steady hand throughout the fourth and final day and almost steered the visitors to an unlikely draw, bringing up his 18th first-class ton with a beautiful straight drive late in the day.
But it wasn’t enough for the Redbacks. Paceman Gabe Bell (3/44) claimed the final wicket to complete a 189-run win and give him seven scalps for the match.
Resuming at 1/16, South Australia lost regular wickets throughout the day.
The visitors managed just 136 in their first innings in reply to Tasmania's 185.
The Tigers pushed further ahead on day three, posting 8/353(dec) headlined by a patient 113 from Player of the Match Jordan Silk and runs to veteran George Bailey.
South Australia sit at the foot of the 2018-19 Sheffield Shield table without a win from five matches this season.
The Redbacks next face New South Wales at the SCG from 7 December in the final round of Shield action before the Big Bash break.