Laurent Blanc's men had been scheduled to take on the Ligue 2 outfit at Stade Francis-Le Ble in the round of 32 on Sunday.
However, the clash will now need to be rearranged after officials opted to call the game off due to concerns over the state of the pitch, with more heavy rain expected over the next 24 hours.
A statement on PSG's official website read: "Following bad weather in Finistere (West Brittany) in recent days, the pitch at the Stade Francis-Le Ble is unplayable.
"The match against Brest originally scheduled for Sunday at 14:15 (local time) has been postponed.
"Because of this postponement, coach Laurent Blanc has decided to allow his players to rest on Sunday. The resumption of training is scheduled for Monday.
"The date of the rearranged match will be announced at a later date."
Ligue 1 leaders PSG will resume their league campaign at Ajaccio on January 11.
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