Oldham's Danny Philliskirk's hit the post with an early piledriver, but County struck after 13 minutes when Ronan Murray wriggled free inside the box and slotted home an angled eight-yard shot.
County went down to 10 men in the 55th minute as Leacock was adjudged to have hauled back Jonson Clarke-Harris just outside the box and was sent off for denying a clear goalscoring opportunity.
Oldham cashed in after 67 minutes, with Philliskirk teeing up the unmarked Dayton to score with a low drive from 10 yards.
And Philliskirk struck again six minutes from time when he collected Dayton's pass 12 yards out and slammed in his second of the night.
Oldham's teenage substitute Jordan Bove was sent off in the closing stages - just two minutes into his debut - for a high tackle on Kieron Freeman.
But Oldham made it five in injury time as Philliskirk's shot was saved and Clarke-Harris tapped in the rebound.
- Sports & Recreation
- Danny Philliskirk
- Dean Leacock
- James Tarkowski
- James Dayton