Brezovan pulled off a series of tremendous saves after half-time and frustrated a Latics team, who were chasing their third successive home win.
Oldham, who have suffered only one league defeat at Boundary Park this season, created the majority of goal scoring chances.
But struggling Swindon earned their first away point under new manager Danny Wilson and could have even snatched victory themselves as French striker Hamdi Razak almost marked his debut with a late goal.
Brezovan was soon in the action when he saved a lob from Oldham's veteran on-loan striker Dean Windass.
There was an opening for the visitors when Gordon Greer skied his shot at the far post after latching on to a corner but Oldham got back on top and Brezovan held Mark Allott's stinging drive.
Swindon full-back Kevin Amankwaah then cleared from Chris Taylor, before home goalkeeper Greg Fleming did well to touch Billy Paynter's shot behind for a corner.
Brezovan also thwarted Lee Hughes before the break as the ex-Premier League poacher tried to add to his 15 goals this season.
Windass, another player with vast experience, could have broken through on 51 minutes only to pull his volley off target after being teed up by Hughes 12 yards out.
The chances began to flow for Oldham in the last 25 minutes, starting when Danny Whitaker found space inside the box but was quickly closed down by Greer.
Hughes headed off target when he should have done better from a set-piece and when he did threaten to open the scoring, Brezovan somehow clawed the ball away from the top corner.
The keeper saved his best for the last five minutes as he dived full-length to thwart Oldham substitute Lewis Alessandra when the youngster looked sure to make it 1-0.
But Oldham keeper Fleming also had to be alert when Razak unleashed a ferocious volley from the edge of the box.