A seventh-minute goal from midfielder Dwayne Mattis extended Bury's winning run in League Two to six games.
Floodlight failure at Gigg Lane twice delayed the game, but the home side made a bright start with sharp attacks.
Mattis fired in his first of the season after a goal-mouth scramble, Andy Bishop's shot ricocheting back to him.
Wrexham's Mike Ingham made fine saves from Bishop, Glynn Hurst and Richie Baker, while Bishop and visitor Maheta Molango had offside goals ruled out.
To add to Wrexham's woes, Ryan Valentine and Jon Newby went off injured, the latter ruled out of Saturday's FA Cup visit of Stevenage.
Bury: Schmeichel, Brass (Woodthorpe 76), Scott, Fitzgerald, Kennedy, Pugh, Mattis, Baker, Buchanan, Bishop, Hurst (Youngs 79).
Subs Not Used: Barry-Murphy, Adams, Speight.
Booked: Mattis, Baker.
Goals: Mattis 7.
Wrexham: Ingham, Valentine (Roche 46), Pejic, Steve Evans, Mike Williams, Mark Jones, Danny Williams, Crowell (Johnson 71), Done, Molango, Smith (Newby 46).
Subs Not Used: Michael Jones, Morgan.
Ref: R Shoebridge (Derbyshire).