County had midfielder Andre Boucaud controversially sent off after 22 minutes for what was deemed a dangerous tackle on Cliff Byrne.
The visitors took the lead just 38 seconds after half-time as Yoann Arquin delivered a right-wing cross and Joss Labadie volleyed home from 10 yards, but Oldham levelled after 49 minutes when debutant Matt Derbyshire reached Lee Croft's through-ball and rounded the goalkeeper to score from 12 yards.
The Latics took command and earned a 69th-minute penalty following Dean Leacock's push on Derbyshire, only for the spotkick to be blazed over by Robbie Simpson.
County regained their lead with a freak goal nine minutes from time as Jamal Campbell-Ryce charged down the right and saw his cross from almost on the touchline sail over goalkeeper Alex Cisak.
Yet Oldham had the final say after 84 minutes, with another debutant, ex-Everton starlet Jose Baxter, netting a close-range volley from Christian Montano's flick.