Robins defender Aden Flint levelled with his first goal since September in a scrappy game ruined by gale-force winds.
Oldham took just five minutes to grab the lead as Korey Smith found Danny Philliskirk, who turned before unleashing a superb 25-yard drive into the top corner.
But City equalised in the 10th minute when Oldham failed to clear Bobby Reid’s right-wing corner and Flint hooked in the loose ball from eight yards.
The visitors suffered an injury blow which saw Marvin Elliott carried off on a stretcher, but Tyrone Barnett forced a smart save by Oldham goalkeeper Mark Oxley.
Bobby Reid and Wes Burns also went close, while Oldham striker Jon Stead curled a long-range free-kick inches off target.
Philliskirk’s back-post header tested City keeper Simon Moore, who was relieved to see a Gary Harkins piledriver flash over the bar.
Robins substitute Jay Emmanuel-Thomas fired a half-chance wide in the last five minutes as both sides settled for a point.