Latics were backed by a 2,073-strong following at the weekend as Matt Smith's header 11 minutes from time condemned the Shakers to relegation.
The away team had an allocation of 2,100 but, despite being notified beforehand that it was possible to pay on the day, around 300 Oldham supporters were turned away.
A statement on Bury's official website read: "Bury Football Club offer our sincere apologies to those supporters who were unable to gain access to the stadium for Saturday's game with Oldham Athletic.
"The club have launched an investigation into the events prior to (the) game.
"Naturally we are extremely disappointed to see supporters turned away and we shall be meeting with the clubs safety department and the police in order to ascertain just what went so badly wrong.
"We shall inform supporters of the outcome of that investigation in due course."
Oldham chairman Simon Corney told the Oldham Chronicle: "It is not right, it is poor. You have to question what has gone on.
"Whether Bury selling tickets on the day was the right thing, or whether police - who are sometimes quick to blame clubs - have done their job properly, I am sure there will be a proper investigation. I am sorry for any trouble it has caused."