McAndrew claimed 6-55 as SA bowled out the visitors for 198 at Karen Rolton Oval on Saturday, before Test star Head helped make light work of the 95-run fourth-innings chase.
Head blasted 58 from 48 balls, including five boundaries and three sixes, to put SA within touching distance of victory.
Fellow opener Henry Hunt (29 not out) was unbeaten as he safely guided SA to 1-96.
The result puts SA second on the Shield table with one round to play, while NSW are now out of contention.
Tasmania, who suffered a 70-run defeat to leaders Victoria, are also now unable to overtake SA.
Ryan Harris' side had a 104-run lead over NSW on the first innings and were in a commanding position when McAndrew and Jordan Buckingham (2-42) ripped through the Blues' top order on day three.
Sam Konstas (28) was out lbw to Buckingham and Lachlan Shaw (64) provided some resistance with his second first-class half-century.
The NSW tail wagged as Charlie Stobo (36) played a part in putting on 62 runs for the final two wickets.
But it was never going to be enough, and Head quickly put any thoughts of a NSW fightback to bed.Â
After ending a 29-year Shield drought last summer, SA take on Victoria in the final round as they seek to book their spot in a second successive final.
SA are eyeing consecutive titles for the first time in the competition's 133-year history.
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