Sayers, displaying masterly ball movement after SA won the toss in overcast conditions, collected his 15th five-wicket innings haul in first-class cricket.
The paceman took 6-55 from 23 overs, claiming the initial four wickets for SA.
The only resistance to Sayers was debutant Daniel Solway who scored a century to guide New South Wales to 7/274 at stumps.
Sayers struck twice early, uprooting the middle stump of Daniel Hughes for a duck and bowling Nick Larkin (16) with an expert off-cutter.
First-drop Solway rallied in a 61-run stand with Moises Henriques, who made 30 but fell just after lunch to another Sayers indipper, trapped lbw.
Sayers added the Blues' other debutant, Matt Gilkes, caught behind for a duck, and Wes Agar picked up Nick Bertus (one) to leave NSW wobbling at 5-107.
Solway combined with veteran Peter Nevill (53) for a valuable 108-run partnership, with the first-gamer racing a quick single to leg to reach his century from 229 balls.
Nick Winter was unfortunate to not snare a wicket through various LBW shouts throughout the day, while Wes Agar (1-50) added the wicket of Nick Bertus (1) with an outside edge to the slips.