Agar continued his impressive start to the season, ending the penultimate day with figures of 3/44 including the wicket of last-gamer George Bailey for nought.
Bailey was Agar's second victim in as many balls when he left a fine delivery that crashed into the off stump, as the Tigers were reduced to 3/28.
The Tasmanian veteran is playing his final first-class game in a 15-year career.
Agar, 22, now sits third on the 2019-20 Shield wicket-taking list with 21, but it was Ben McDermott who halted South Australia's charge with an unbeaten 88 from 232 balls.
The Tigers' second innings will resume on Tuesday at 6/241, an overall lead of 149 runs with McDermott and young bowler Lawrence Neil-Smith (31*) at the crease.
Agar also ended the resistance of opposition skipper Jordan Silk on 29 (118), while fellow quicks Nick Winter, Chadd Sayers, and Joe Mennie all have a wicket each.
Middle-order batsman Jake Doran then dug in with McDermott for a 94-run fifth-wicket partnership before Mennie made Doran miscue a pull shot.
Sayers claimed the wicket of first-innings centurion Simon Milenko (16) with just his fifth delivery with the new ball, and SA's veteran swingman will now take it into the final day 19 overs old.
Opener Jake Weatherald starred with 126 in the Redbacks' first innings of 346, with skipper Alex Carey next best with 73.