The home side took the lead after just nine minutes as Gael Bigirimana robbed Joe Crowe of possession and John McGovern latched onto the loose ball, feeding Andrew Mitchell who was able to convert from close range.
Dungannon doubled their lead in the 20th minute after Kealan Dillon’s corner was cleared, but only as far as Leo Alves, who took a clever touch before slotting into the bottom right-hand corner from outside the box.
Adam Glenny added a third before the break after great build-up play from the Swifts.
The move started when Bigirimana controlled a mishit clearance from Carrick stopper Ross Glendinning and found McGovern.
The forward then played a clever one-two with Bobby-Jack McAleese before sending the ball across the box for Glenny to finish.
It was the visitors who made the brighter start to the second half and netted on 57 minutes when Paul Heatley slotted home substitute Danny Gibson’s deflected cross from the left wing.
Both teams continued to push for goals with Dillon firing just over the crossbar for Dungannon and Declan Dunne doing well to clear the ball away from the onrushing Gibson, who was through on goal.
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