Brezec grew up in Sežana, where he also made his professional debut with the local team Kraski Zidar. During his high school years he was a member of the basketball team and a major star of the ŠKL League, the national high school league. As a professional, he played for the Slovenian Premier A League club Union Olimpija Ljubljana from 1998 until 2001.
Brezec spent the next 7 seasons playing in the NBA in the United States and Canada. He returned to Europe on July 20, 2008, when he signed a 2-year contract with Lottomatica Roma of the Italian league.
Brezec was chosen by the Indiana Pacers with the 27th overall pick of the 2000 NBA Draft. After he spent three years in Indiana with little playing time, he was picked up in the 2004 NBA Expansion Draft by the Charlotte Bobcats. As the Bobcats' starting center, Brezec averaged career-highs of 13.0 points and 7.4 rebounds per game during the 2004-05 NBA season. He also scored the first points, made the first turnover, and hit the first free-throws in the history of the franchise.
At the 2008 NBA trade deadline, which was on February 21, Brezec, along with cash considerations, was traded to the Toronto Raptors in exchange for Juan Dixon. During his Raptors debut, he hit all five of his shots and finished with 11 points, three rebounds and a block in 13 minutes off the bench against the New York Knicks in a February 24 win.