Set 298 runs to win, the Redbacks were bowled out for 201 on Monday's final day, Tom Cooper top-scoring with 77 and captain Travis Head making 29.
Sayers continued his remarkable match - after taking career-best bowling figures of 13/131, he cracked 47 runs in the Redbacks' second innings.
Hazlewood took 6-35 from 18.4 overs as the Blues extended their lead atop the Shield table with a third consecutive win.
Seamer Trent Copeland claimed 3-50 and Test spinner Nathan Lyon returned 1-50 from 15 overs.
SA resumed at 1-1 on Monday and promptly lost two wickets in a six-ball span when nightwatchman Nick Winter (six) and opener Henry Hunt (six) were dismissed in the opening 20 minutes.
Redbacks skipper Head, after making 109 in the first innings, looked in fine touch and raced to 29 from 32 balls before falling.
The left-hander launched a drive and edged to first slip from the bowling of Test spinner Lyon who found turn in his early spell on a fourth day Adelaide Oval pitch.
SA stalwart Callum Ferguson made 25 from 97 balls before nicking a short, wide delivery from Hazlewood to wicketkeeper Peter Nevill and when Harry Nielsen (duck) and Tom Andrews (four) fell, the Redbacks were reeling at 7-107.
Sayers then provided some lower-order spark, smacking six fours and a six, in a rapid 94-run partnership with Cooper.
The Redbacks next play Tasmania at Adelaide Oval from Monday, 11 November.