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A minute's applause was held before kick off, celebrating the life of Ashington supporter Craig Dickinson, who died just days before the game.
Jamie Barkway was back to captain the side after a spell at Pickering, where he was getting game time following a suspension.
Mutty made his competitive home debut, after recovering from a cracked rib at Tadcaster in pre-season.
Luke Aldrich missed out after taking a knock at Matlock in midweek.
Mutty opened the scoring on nineteen minutes, driving a low shot beyond the keeper.
The young wide player had been a constant thorn in Ashingtons side early on, resulting in left back Rory Powell being booked in a rash attempt to halt him.
Rinor Ejupi doubled Garforth's lead midway through the second half with a half volley that rattled the far stanchion of the Ashington net.
Garforth had to ride their luck at times, as the away side missed an open goal and were denied by a couple of goal line clearances.
As Ashington committed more players forward, they were killed off with a swift break that resulted in George Harrison netting a third.
Garforth will travel to Runcorn Linnets in the next round, but for now, all eyes will be on the Bank Holiday Monday clash at Emley.