The Bees took the lead just before the break when Sam Saunders’ mazy run was ended with a shove by Andy Holdsworth in the penalty area and Kevin O’Connor made no mistake from the spot.
Oldham levelled seven minutes after the restart through a superb 25-yard curler from the edge of the box from the right foot of Blackman.
The home side had the better of the second-half chances with Carl Cort seeing a header tipped over and Steve Kabba slicing wide when well placed.
But the game was marred by a string of bizarre decisions against both sides from referee Grant Hegley.
The closest the visitors came to snatching the points was when Pawel Abbott missed a glorious chance with 10 minutes to go when he headed wide with the goal gaping.
Brentford midfielder David Hunt flashed a half-volley narrowly over in the closing minutes.