Joseph Kenny netted Liversedge’s other goal.
Joining them in the next round will be Hemsworth Miners’ Welfare following their 4-0 victory at Kendal Town. Jason Yates opened the scoring on 15 minutes before further goals from Luke Danville, Adam Knight and Jason Davies sealed the victory.
A goal seven minutes from time by Danny Edwards earned Nostell Miners Welfare a replay after a 2-2 draw with Stockton Town.
Adam Hayton netted Nostell’s opening goal.
Garforth Town also face a replay after their match at Shildon ended goalless.
Knaresborough Town and Yorkshire Amateur both exited the competition after respective 3-0 defeats to Newcastle Benfield and Billingham Town.
Harrogate Railway suffered the heaviest defeat of the weekend, going down 10-1 at home to Whickham.
Meanwhile in Division One of the Northern Counties East League, Selby Town moved into fifth place thanks to a 3-2 home win over Dronfield Town.
Ryan Gothard and a Christopher Jackson penalty gave Town an early 2-0 lead only for the visitors to draw level through Daniel Wood and Stuart Hill. But Liam Flanagan’s goal 10 minutes from time proved to be the winner for Selby.
Glasshoughton Welfare are still seeking their first point of the season after a last-gasp 3-2 defeat at Parkgate. Welfare’s goals came from Craig Tonkinson and Lewis Pickering.