County dominated the opening exchanges and created numerous half chances, the best falling to Alan Judge in the 13th minute.
The diminutive midfielder was slipped through by Andre Boucard but after taking the ball around Dean Bouzanis in the Oldham goal, he could only find the side netting from a tight angle.
The home side deservedly took the lead though in the 35th minute when a brilliant pass behind the Oldham defence by Boucard played in Jamal Campbell-Ryce who slid the ball across for Showunmi to tap home.
The Latics came out stronger in the second half and looked to have levelled in the 57th minute when striker Matt Smith beat County stopper Bartosz Bialkowski to a high ball but his goal-bound header was cleared off the line superbly by Gary Liddle.
County could have wrapped things up with 12 minutes to go when Yoann Arquin pulled back for Judge who sliced wide but Showunmi's first-half goal proved decisive.