The goalkeeper began his career in his native Slovakia with PS Bratislava followed by moves to Vinohrady Bratislava, SKP Devín, Slovan Břeclav, 1. HFK Olomouc before joining 1. FC Brno. Brezovan played for FK Inter Bratislava on loan before he was spotted by Swindon Town boss Dennis Wise during a trial at Crewe Alexandra. Wise signed the Slovak on a season long loan with view to make the deal permanent later.
His Swindon debut came on the opening day of the 2006/07 campaign - away at Hartlepool United, Brezovan saved two Hartlepool penalties from Joel Porter and Ritchie Humphreys. Swindon finished 1-0 winners.
Following his heroics in the game against Hartlepool and a string of impressive displays, Peter was awarded the PFA Player of the Month award for September .
Falling awkwardly in the 76th minute of the home game against Grimsby Town on the 14 October 2006, which Swindon won 3-0, Brezovan suffered a broken arm.
On 29 March 2007 it was confirmed that Brezovan had signed a two-year contract with Swindon.
In the 2007-2008 season he was confirmed as second string keeper, after Phil Smith did well in goal in his first few games for Town in League One. Brezovan was given number 12 and slumped in the Town rating after conceding a goal in a 4-1 win at home to Brentford
Mid-way through the 2007/2008 season, first choice goalkeeper for Swindon Town Phil Smith was injured in a match at home to Port Vale. Brezovan was granted a first team slot for the next few matches.
In September 2007 Brezovan spent two days on trial with Everton, and he was being trialled by Bristol City but unfortunately for him his chances were done a great deal of harm when he allowed a speculative, long-range shot in through his arms during their 4-0 defeat by Ajax in a pre-season friendly on 15 July 2009.
In October 2009 he was on trial at Crewe Alexandra but was not offered a contract.
He was released by Swindon Town in the summer of 2009.
In December 2009 Brezovan signed a short-term deal with Brighton and Hove Albion keeping him at the club until January 2010. He made an immediate debut on 5 December in a 1-0 away win against Exeter City; a game in which he also saved a penalty.