Wythenshawe Town advanced to the next round thanks to an Isaac Abankwah goal early in the second half.
A cagey first period saw little in the way of goalmouth action, although George Boyds’ second minute shot almost caught the Miners cold.
Both physios were in the thick of the action with players on either side needing treatment as the half stuttered to a stalemate.
The away side again started quickly and Abankwah capitalised on space at the back post to head them into the lead on fifty minutes.
The game opened up and although Garforth had opportunities to equalise through chances created by Jordan Emery and Jay Davis, Wythenshawe bodies were on the line when needed.
The visitors were able to see out five minutes of added time to earn their place in the next round.