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First-half goals from Leroy Millar and Kevin O'Hara set the Invermen on their way, before Matt Ridley added a third after the break.
The win relieves some pressure on manager Gary Haveron after last week's disappointing 4-2 defeat by Cliftonville.
For Dungannon it is back to the drawing board as they lacked a cutting edge in the box and have now won just one of their past 10 league fixtures.
It was an exciting start for Larne and, with just two minutes on the clock, Millar came close to opening the scoring when his left-footed effort from 20 yards flashed just over the crossbar.
On 23 minutes, Larne went close once more as Millar turned provider with his low delivery from the left met by O'Hara, who tried a clever back-heel that almost caught Dungannon unaware, but stopper Declan Dunne got down well to make the save.
The opener did arrive six minutes later for the home side when Millar again threatened down the left, before he played a smart one-two with Sean Graham and curled a low right-footed effort into the bottom right-hand corner of the net.
Just four minutes after, Larne doubled their advantage after the Swifts' Jamie Glackin played a sloppy ball across his defence which was intercepted by O'Hara who unleashed a right-footed shot from 20 yards that flew into the left-hand corner of the goal giving Dunne no chance.
Dunagannon started the second period at pace and almost grabbed a goal back within 90 seconds when a cross from the right was fired towards goal from close range by Andrew Mitchell, but Rohan Ferguson made a good save to deny the effort.
The Swifts were to be punished for not taking that chance on 58 minutes, however, after Graham swung in a corner from the right and Ridley rose highest to power his header past Dunne and into the net.
Five minutes from time, Leo Alves almost pulled one back for the Swifts but the substitute's superb free-kick from 25 yards was superbly tipped around the post by Ferguson to ensure yet another Larne clean sheet.
"I thought we took our goals well, I felt our shape was good and we caught them out.
"Really pleased to see Matt Ridley score, he gets on so many at the other end of the pitch to clear and has just been unlucky hasn't scored more."
"But we have to give a better account of ourselves without the ball than we did tonight.
"We gifted Larne three goals. The first one was a poor header, the second one was an awful ball and the third one came about from playing ourselves into trouble.
"I'm frustrated overall."