He produced two penalty saves to foil a dominant Hartlepool, who also hit the woodwork twice.
Brezovan set the pattern for his first half when he saved Joel Porter's third minute penalty after the home striker had been fouled by Ricky Shakes.
Hartlepool paid the penalty for the miss when the unmarked Lee Peaock headed Swindon in front from an 11th minute corner.
But the home team kept on pressing forward to create a string of chances.
Porter's low 20-yarder was deflected into the air right in front of Brezovan, but the big keeper somehow still managed to save and then did the same at the feet of Lee Bullock at the expense of a corner.
Matty Robson and Porter both sent long-range shots wide of the target, and just before the break Brezovan kept out Porter's header as it seemed to be going for the top corner of the net.
His best save came two minutes before the break when he managed to stop a close-range volley from Michael Nelson after another home corner.
In the second half it was the woodwork that came to Swindon's rescue when Brezovan was beaten at last by the home team.
After 66 minutes Lee Bullock's header came back off the bar after a great right-wing cross from James Brown.
And as the pressure mounted, on 69 minutes it was Porter who hit the woodwork, his glancing header coming back into play off the inside of the post after a left-wing cross from substitute Eifion Williams.
But Brezovan wasn't finished yet, and his second spot-kick save came after 70 minutes. Porter was fouled again, this time by Jack Smith, and when skipper Richie Humphreys stepped forward to take the kick, the keeper threw himself to his left to beat out the shot.
That latest save knocked the stuffing out of Hartlepool, and although they stayed on top for the rest of the game, they couldn't create another real chance.