The away side should have taken an early lead with just two minutes on the clock. Dungannon failed to clear their lines but Liam McStravick, seizing on the loose ball, could only drag his first-time volley inches wide of the far post.
The Reds threatened again when Hale slipped Ben Quinn through on the left, only for Declan Dunne to react smartly to turn the winger's low effort around the near post.
That warning went unheeded and Cliftonville broke the deadlock on 25 minutes when Aaron Wightman whipped an inviting delivery into the six-yard box from the right flank for Healy to steer past Dunne on the stretch.
Magilton's men doubled their advantage just three minutes later, after Quinn's overhit cross was rescued on the right by McStravick, who teed up Hale to squeeze a low finish past Dunne at his near post.
The hosts halved the deficit on 70 minutes with a move bearing a striking resemblance to Healy's earlier opener.
Corey Smith picked out Gordon at the back post and he teed up former Everton man McAllister for a simple close-range tap-in.
Any hopes of a comeback were swiftly extinguished moments later, however, as McStravick unlocked the Swifts' backline with a sumptuous through ball for Healy, who rifled an emphatic finish past Dunne to restore the away side's two-goal cushion with his fifth of the campaign.