Blues openers Daniel Hughes and Nick Larkin both made centuries for the home side while Moises Henriques scored 99 as NSW claimed the ascendancy.
Young Redback Conor McInerney earned his maiden first-class wicket on the stroke of tea to dismiss Henriques caught and bowled in one of the highlights for the visitors.
Stand-in captain Jake Lehmann also claimed multiple wickets for the first time in a Shield match, picking up captain Peter Nevill (57) and Steve O'Keefe (29) late in the Blues’ innings.
Leg-spinner Adam Zampa and part-time spinner Tom Cooper both earned two wickets apiece for South Australia.
New South Wales passed SA’s total of 380 after lunch then increased the run rate in the afternoon before declaring just before the end of the day’s play.
At stumps, the Redbacks were 0/18, still needing 143 runs to make NSW bat again.
Openers Jake Weatherald (5*) and McInerney (11*) will resume for South Australia on Monday.
SA swing bowler Daniel Worrall did not bowl on the third day of play and is being assessed on the recurrence of a back problem. He is unlikely to bowl again this match.