Stuart Vasey got the winner for Neil Crowe’s side a minute before half-time.
Elsewhere in the Trophy, Lancaster City were held to a 1-1 draw by visitors Droylsden.
Jacob Gregory gave the Dolly Blues a fifth-minute lead but Joshua Heaton earned the Bloods a replay with a 34th-minute equaliser.
Burscough went out, losing 2-1 at home to Kendal Town.
All the goals came late on as Oliver Wood (73) and Robert Bowman (85) put Town in command.
Dean Shacklock (88) got a consolation goal.
And Clitheroe crashed out of the FA Trophy, hammered 5-0 at home by Stocksbridge Park Steels.
AFC Fylde extended their unbeaten start to the season to 13 games with a thrilling 3-3 draw against Kidderminster Harriers in the National League North.
The hosts twice led the game, but needed a late equaliser to cancel out a stunning Danny Rowe strike and James Hardy’s double.
The hosts took a surprise lead in the 13th minute when Arthur Gnahoua netted.
The Coasters drew level when Rowe let fly from 25 yards but moments later the home side were back ahead through Elton Ngwatala.
The Coasters equalised when Rowe’s initial volley was saved, but Hardy put in the rebound and then put them ahead in the 72nd minute.
But Harriers’ Keith Lowe scored from close range late on to level the scores again.
After an embarrassing drubbing at the hands of Solihull Moors on Tuesday night, Southport completed a miserable week with a 2-1 defeat at Guiseley.
The hosts thus overtook the Sandgrounders in the National League table, leaving them in bottom position on goal difference.
The opening goal came four minutes into the second half, Jake Cassidy converting Jermaine Hylton’s pass, but that was cancelled out by Jamie Allen’s shot in the 70th minute.
Port looked the most likely to snatch a win but when Mike Rankine headed in Adam Boyes’ cross it was all over for them.