Brane, as he is often known, started playing football at the youth team of Svoboda Ljubljana. In 1965 he went to Ljubljana's more famous club, Olimpija. He made his debut on 20 May 1966 against Partizan in Belgrade, where he scored both Olimpija's goals for a 1–2 win. He stayed with Olimpija until 1973. During that time he played 181 matches and scored 33 goals.
In 1973, he moved to Hajduk Split and stayed there for two seasons. In both seasons Hajduk won the national champion and cup winner titles. In 1975 he signed a two-year contract with Schalke 04 of the German Bundesliga for what was a world record fee for that season. In the 1976–77 season, Oblak's second year at the club, Schalke 04 finished the league in second place, behind Borussia Mönchengladbach. In the summer of 1977, Oblak went to Bayern Munich. He stayed there for three seasons and in his final season with the club (1979–80), Bayern became the German champion. After that, Oblak played in various Austrian amateur clubs until 1987, when he retired as a football player.
Oblak's debut for the Yugoslav national team came in a friendly match against Romania in 1971. The same year he had the honour of playing at Pelé's farewell match at the Maracanã in Rio de Janeiro. At the 1974 World Cup in Germany, he and his teammate Danilo Popivoda became the first Slovenians to play in a World Cup. Even more impressive is that Oblak was selected into the best squad of the World Cup. In 1976 he played at the European Championship, which was hosted by Yugoslavia. His excellent pass for the first goal in the semi final match against Germany is still remembered among football experts. After the tournament, he retired from international football. In altogether 46 matches he scored six goals.
After retirement as a football player, Oblak started coaching at various Slovenian clubs, including three times at Olimpija Ljubljana,NK Naklo and NK Koper. He proved to be a good youth coach, so in February 2004 he was named the new Slovenia Under-21 national coach. In May 2004, Slovenian national team head coach Bojan Prašnikar was fired due to bad results and on 24 May 2004 Oblak was chosen to replace him. The team started brightly under his guidance, with the highlight being a 1–0 over Italy (eventual World Champions) in a World Cup qualifying match. However, the team soon ran into a series of bad results, and the pressure mounted on Oblak, until finally in late 2006, he was fired from the job. He was succeeded by Matjaž Kek. In 2009 he again took the managers job at his home club NK Olimpija Ljubljana. He was relieved of his duties after only four games.