Both sides had half-chances in the first half before Adamson's overhead kick put the Swifts ahead three minutes before the break.
Michael Ward set up Adamson's second goal on 47 and Ryan McIlmoyle was the provider for the third on 67.
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Ward had the first chance of the match in the seventh minute but Kyle McVey averted the danger by making a block.
Adamson almost opened the scoring in the 16th minute but he headed McIlmoyle's cross over the top.
Coleraine's Gareth Harkin fired a free-kick just wide in the 25th minute but Dungannon continued to press and Davy O'Hare saved well from Dermot McCaffrey.
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However, Adamson's overhead kick put the Swifts ahead three minutes before the break and they never really looked back in the contest.
Ward's unselfish pull-back set up Adamson's second goal two minutes after the break and any slim prospect of a Coleraine comeback was ended when the striker completed his hat-trick 20 minutes later.