The team currently lying ninth in the Highland League had thrashed league rivals Rothes 8-0 before edging out Turriff United 3-1 after extra time in the previous rounds.
Brechin are seven points clear at the top of the Highland League and take on the team currently bottom of the top flight after beating Newtongrange Star 1-0, Huntly 3-2 and Cowdenbeath 4-1 after extra time.
Meanwhile, Premier Sports will broadcast live two all-Premiership ties and one between two Championship rivals.
Cup holders Celtic entertain Kilmarnock on Saturday, 18 January, Falkirk host Raith Rovers the following noon, while the derby between Dundee and Dundee United completes the fourth-round fixtures on the Monday evening.