The Jets had looked poised to walk away with the victory after rising A-League Men star Lachlan Rose returned to haunt his former side with a goal and an assist at Campbelltown Stadium on Friday night.
Rose scored in the second minute before Eli Adams built Newcastle's lead in the seventh.
But Sawyer, injected into the match in the 56th minute, refused to throw in the towel and scored six minutes later to spark a second-half challenge.
Sawyer had been fouled by Aleksandar Susnjar, giving a free kick to fellow substitute Saif-Eddine Khaoui.
Khaoui proved his class with a quality cross, with Sawyer finishing the play off with a headed goal.
The Jets restored their two-goal lead when Rose popped up again in the 71st minute to set up Adams' second goal of the night, before Bulls defender Tomi Uskok returned serve in the 82nd-minute.
After a number of missed opportunities, Sawyer broke through with another header five minutes into added time.
While the Jets will walk away disappointed, the draw at least keeps both sides' slim finals hopes alive.
The Jets are seven points off sixth-placed Adelaide, sitting ninth in the league on 28 points.
Macarthur are eighth on 29 points with five rounds remaining.
Milestone man Luke Brattan had tried to take matters into his own hands in the 25th minute with a shot from just outside the box but sent the ball flying above the crossbar.
In his 22nd league game for Macarthur after moving from Sydney FC, Bulls captain Brattan became just the 12th player to reach 300 ALM appearances.
He did so as goalkeeper Alexander Robinson made his league debut, with No.1 shotstopper Filip Kurto out with illness.Â
A foul by Dane Ingham in the 31st minute gave Brattan another goal opportunity, conceding a free kick within scoring distance.
But Brattan sent the ball straight into the wall of Jets defenders and fluffed the rebound.
18°C