Alfie Rutherford scored in both halves to send Wanderers through however the home side had their chances throughout the 90 minutes to make life more difficult for the National League side.
After a quiet opening ten minutes where both sets of players adjusted to the challenging conditions there were chances for both sides to take the lead within a minute of each other.
Firstly Jesse Starkey send in an inswinging corner that found Aarran Racine in space at the back post but the captain could only stab his effort into the side netting.
Seconds later, at the other end the ball fell for Matt Briggs on the edge of the area but he dragged his effort wide of Harrison Male’s post.
Rutherford broke the deadlock in the 38th minute with the game’s first bit of real quality, cutting inside from the left and flashing a great strike past the helpless Male.
The home side came out fighting in the second half, presentable chances for Cox and Armstrong came and went in quick succession, before Racine’s header was cleared off the line by opposing captain Barry Fuller.
Despite those chances Wanderers extended their leads shortly after the hour mark when Matt Briggs beat Male to a through ball, leaving Rutherford with a simple tap in for his and the away side’s second of the game.
The home side continued to create chances, Racine again proving dangerous from corners, this time heading over then Reece Meekums drew a good save from Stuart Nelson.
As they continued to push Wanderers found space at the other end and nearly added a third when Briggs robbed Cameron Tutt and teed up substitute Jason Prior who’s effort was well saved by Male.
The higher division side used their experience and saw out the final ten minutes fairly comfortably to progress to the next round leaving Adam Hinshelwood’s men to concentrate on their pursuit to join their opponents in the National League next season.