North End made the early chances, with David Amoo's shot deflected wide and Stuart Beavon firing over on the turn. But Oldham went ahead after 27 minutes when Cristian Montano was fouled by Jack King and the resulting 25-yarder was driven home by Baxter, via keeper Thorsten Stuckmann's fingertips.
Baxter almost added a second as his volley through a crowd of players was clawed away by Stuckman, who also denied Matt Derbyshire.
And former Everton youngster Baxter was again involved when Oldham doubled their lead shortly before the hour, finding Jonathan Grounds for a through-ball which Montano drilled home from 16 yards.
It was 3-0 after 69 minutes, Dean Furman seizing on a defensive blunder to slot beyond Stuckman from 12 yards.
In the 79th minute Preston had Shane Cansdel-Sherriff sent off for his second bookable offence, a trip on Lee Croft. North End scored a consolation in the closing seconds of stoppage time when Akpo Sodje shot in from close range after a mistake by Grounds.