The Yeltz scored twice in injury time to overcome Prescot and set up the tie away to the National League North high-flyers.
Spennymoor are currently fourth in the table – the second tier of non-league and one level above Halesowen.
Kidderminster Harriers, also of National North, have been handed a home tie against league rivals York City.
The matches will be played on Saturday, November 24, with the winners receiving £5,000 in prize money.
National Premier clubs – including the likes of Wrexham and Leyton Orient – enter at the following stage, the first round proper.