Barry Gray departed as Cliftonville boss after Dungannon inflicted a sixth defeat in seven games.
A hat-trick from Daniel Hughes, a former Reds player, saw the Swifts claim a convincing win at Stangmore Park.
Gray had been under mounting pressure in recent weeks and this loss proved the final straw.
In a statement issued shortly after the full-time whistle, the club confirmed it had “parted company” with him.
His final game was one to forget, with the Reds outplayed for most of the match.
They fell behind in the 30th minute through Hughes, who spent two years with Cliftonville, but is now showing his quality at Dungannon.
He beat the offside trap to collect a pass and close on goal before finishing confidently into the top corner.
Hughes was the Swifts’ biggest threat, and nearly made it 2-0 after meeting Oisin Smyth’s cross early in the second half.
Cliftonville, though, briefly rallied and managed to get themselves back in the game.
In the 57th minute a corner from Ross Lavery was crossed back in by Joe Gorman, and Jamie Harney rose to head home.
But Dungannon soon regained the attacking initiative, with Hughes to the fore again.
He put the Swifts back in front five minutes later, taking a pass from Smyth on his chest before firing home.
And Hughes completed his hat-trick in the 66th minute.
He timed his run to meet Paul McElroy’s cross, dashing into the six yard box to score with a downward header.
The Swifts, who gave a debut to new signing James Ferrin as a second half sub, continued to push for goals.
Hughes had a header tipped over, while a deflected strike from sub Rhyss Campbell was pushed wide.
McElroy was denied at the near post after a mazy run, while Smyth had a shot blocked late on.