Institute kept up their push for a place in the JJB Sports Premiership top six by beating the basement club.
Dungannon had the better chances in the first half and Ryan McIlmoyle struck the underside of the crossbar.
Paul Brown put Institute ahead on 52 minutes with a header from Eamonn Seydak's cross.
Kevin Ramsey lobbed keeper Alvin Rouse eight minutes later to make it 2-0 and Declan Divin rifled in from 20 yards for the third after 77 minutes.
Institute's biggest league win of the season takes them to within two points of sixth-placed Cliftonville, who lost 4-2 at Crusaders.
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