[11] Puc was first noticed by RK Borac in 1983 when their goalkeeper, Yugoslav international Zlatan Arnautović, spotted him and reported his findings to the club officials.

Abas Arslanagić, the coach of Borac, which was one of the top Yugoslav clubs at the time, wanted the young teenager to join his team immediately, however, Puc was persuaded by Miro Požun to finish high school first and complete at least some form of education.

[10] Puc listened to his mentor, finished high school one year later and finally joined Borac in 1985 where he signed his first professional contract.

[13] Miro Požun was the head coach of Celje during the 1994–95 season with whom Puc won his first Slovenian league and cup title.

[13] His first taste of international success came at the 1987 Junior World Championship when Yugoslavia won gold, and Puc was noticed as the most prominent player of the winning team.

One year later he won a bronze medal with the Yugoslav senior team at the 1988 Summer Olympics in Seoul.

[13] Soon afterwards he moved to Florida together with his wife Jasenka, who is the daughter of the Croatian handball player and 1972 Olympic gold medalist Hrvoje Horvat,[2] in support of their son Borut's tennis career.

He and his wife sold all of their family possessions and enrolled their son to the Nick Bollettieri Tennis Academy.

[21] At the time of his father's death, Borut Puc was ranked 502nd on the ATP list[22] with Goran Ivanišević as his tennis coach.

[2] In early 2011, Puc was diagnosed with lung cancer that spread to his liver and bones,[4] and succumbed to the disease on 19 October 2011 in a San Diego hospital,[2] just a few days before the Champions League game between Zagreb and Barcelona, with the revenue of the match intended to be donated to help cover the costs for his treatment.

He is survived by his wife Jasenka and son Borut, a tennis player who resides in Florida, where the family moved to in 2005.

A clip of his match-winning goal for Zagreb in the 1993 EHF Champions League final was shown and 15,000 people joined in a minute-long standing ovation in his memory.