South Australia were bowled out for 316 midway through the fourth day with opening batsman Mackenzie Harvey scoring 90.
Shield champions for the past two seasons, South Australia looked well placed to score the mammoth 422 for victory when they resumed on the last day at 1-182. Harvey was unbeaten on 90 and captain Nathan McSweeney 57.
However Mackenzie lasted just 11 balls without adding to his overnight tally before being bowled between bat and pad by Ashton Agar and on 66 McSweeney pushed tentatively at a ball from Kieran Elliott to be well caught low at second slip by Cameron Bancroft.
McSweeney missed an ideal opportunity to continue his push for a Test recall after a strong start to the season. He made a century for Australia A against India A in India early this month then came home to make another one day century for South Australia before passing 50 in each of South Australia's previous two Shield matches.
South Australia lost 9-132 to suffer a second Shield loss in 10 days after going down to Queensland by six wickets, also at Adelaide Oval. South Australia began their title defence well with an innings victory over Victoria at the Junction Oval in the first game of the season.
South Australian coach Ryan Harris believes his side missed an opportunity on Tuesday.
"We had it set up for a good day today," Harris said, before describing the batting as "not good enough." after being bowled out for 154 in the first innings and collapsing again in the second.
During their second innings six of South Australia's top seven batsmen made starts but none cashed in with a big score.
Agar finished the innings with figures of 3-62 while Elliott, Jhye Richardson and Corey Rocchiccioli all took two wickets.
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