Žungul began his career with Hajduk in his native Yugoslavia before controversially moving to the United States where he became a dominant indoor soccer striker.
His mother Danica quickly remarried and moved to Kaštel Lukšić near Split, PR Croatia where baby Slaviša was given his stepfather's last name.
At Hajduk he quickly established himself as the focal point of head coach Tomislav Ivić's setup as well as becoming a fan favourite for his off-the-cuff ways and jet-set lifestyle, often seen in bars around the city while dating models, pinups, singers, and pageant contestants.
While twenty-four-year-old Žungul was years away from even considering a move abroad, he was on increasingly bad terms with Hajduk management—led by club president Tito Kirigin [hr]—over unpaid wages.
On 3 December 1978, Žungul played in Hajduk's last league match before the 1978-79 mid-season winter break—a 5–0 home thrashing of FK Sarajevo at the Stari plac Stadium—scoring twice thus increasing his half-season scoring tally to 12 goals.
[6] He then asked the club management to allow him to travel to New York City for a few weeks in order to accompany his singer girlfriend Moni Kovačič (famous for appearing in Start, the Yugoslav counterpart to Playboy)[7] while further disclosing to them that while in the U.S. he would also be taking part in a few exhibition indoor soccer matches in order to stay in competitive shape for the season restart.
[7] In actuality, through countryman Dragan Popović, Žungul (represented by sports agent Ante "Bekin" Kuzmić) had already agreed a contract with the newly established, Popović-coached New York Arrows of the also recently launched Major Indoor Soccer League (MISL).
[7] Within weeks, in late December 1978, his defection and deception became known back home, creating one of the biggest scandals in Yugoslav sporting history.
Hajduk management was furious, starting a smear campaign against the player in Yugoslav media, branding him a traitor, a deserter, and a drunk.
Though financially well compensated for his services, the move to indoor in professional terms meant being reduced to playing with and against players whose skills and abilities were far inferior to his.
Making his debut in the Arrows' opening game of the season in Nassau Coliseum he immediately established his scoring credentials, recording four goals.
The Arrows responded by trading Steve Zungul to the Golden Bay Earthquakes of the North American Soccer League (NASL), who were playing the 1982–83 MISL season as a guest team, for Gary Etherington and Gordon Hill.
[8] It wasn't, due to the Earthquakes refusing to honour it by referencing the US Supreme Court decision, meaning the player would get his chance in the outdoor game again following a four-year absence.
Žungul made his debut for Yugoslavia in a September 1974 friendly match against Italy, coming on as a 46th-minute substitute for Danilo Popivoda, and earned a total of 14 caps, scoring no goals.