Chasing 298 to win, the Redbacks are 1/1 at stumps on Sunday's third day of play after Sayers had taken 13 wickets across the Blues' two innings.
Sayers' match figures of 13/131 are the seventh-best for South Australia in Shield history.
After taking 8-64 in the first dig, Sayers on Sunday claimed 5/67 as the Blues were bowled out for 253 in their second innings with Nick Larkin top-scoring with 91.
Sayers took the initial two NSW wickets on Sunday before Larkin featured in a steadying 70-run partnership with Moises Henriques, who was dismissed for 38 when well set.
Larkin put on another handy stand with debutant Matt Gilkes (30), adding 60 runs and taking the visitors to a commanding 3-184.
The Redbacks appeared on the ropes but were revived by Sayers, with support from Wes Agar (2/45) and Luke Robins (2/49), as the Blues lost their last seven wickets for 69 runs.
NSW, seeking a third-consecutive victory win to open their Shield campaign, made 289 in their first innings before bowling SA out for 245.
Seamers Harry Conway and Trent Copeland claimed three wickets, spinner Nathan Lyon returned 2-45 from 22.2 overs and his Test teammate Josh Hazlewood (1/60 from 24 overs) went largely unrewarded.
SA skipper Travis Head top-scored with 109 but Tom Cooper (37) was the only other batsman to pass 20.