Tibor Molnar was born in Budapest, Hungary on June 6, 1952. He started his career with Hungarian second-division club Vasas Budapest.
Tibor defected from the communist-led Hungary at the age of 18. After spending some time in a refugee camp in Vienna, Austria, he came to the United States in October 1970.
In May 1971 he made his debut for Dallas Tornado in the North American Soccer League. Tibor played in three league games for Tornado during the season and scored a goal when the Tornado’s won the NASL Championship Series.
Tibor then returned to Europe where he tried to get a contract with the Beerschot club of Antwerp in Belgium. His return to Europe got the attention of the Hungarian Football Federation and he ended up being suspended by FIFA for two years.
In 1973 Tibor played for Toronto Hungaria in the National Soccer League.
He started the 1974 NSL season with the Canadian Hungarians (formerly Toronto Hungaria), but later in the season moved to Toronto Italia. He helped Italia win the 1974 NSL Cup.
During the winters he played for the New York Ukrainians and the New York Hungarians in the German-American League.
Tibor helped Toronto Italia win the 1975 National Soccer League championship with Toronto Italia. During the season he scored 17 goals.
Tibor joined North American Soccer League club Rochester Lancers in February 1976. After four North American Soccer League games for the Lancers he was purchased on June 8, 1976 by San Jose Earthquakes. He played in 13 North American Soccer League games and had three assists for the ‘Quakes in the 1976 NASL season.
In the 1977 North American Soccer League season Tibor played in San Jose Earthquakes, playing in 19 NASL games for the Quakes.
On January 20, 1978 Tibor was traded to the new NASL-franchise Tulsa Roughnecks for Victor Kodelja and future draft choices. Despite playing in four North American Soccer League Indoor games, he failed to settle terms with the Roughnecks. Tibor instead returned to the National Soccer League, where he joined Montreal Castors and helped the Castors win the 1978 National Soccer League championship.
In November 1978 Tibor joined New York Arrows in the newly-created Major Indoor Soccer League. He scored 15 goals in 20 MISL games during the 1978-1979 season. He followed up with 13 goals in 32 games in the 1979-1980 MISL season. In both his seasons with the Arrows the club won the MISL league championship.
On June 25, 1980 Tibor signed for Major Indoor Soccer League club Hartford Hellions. After playing in 22 league and scoring five goals, he joined MISL club Baltimore Blast.
Tibor signed for Pittsburgh Spirit of the Major Indoor Soccer League in September 1981. Despite struggling with injuries he scored 6 goals and had 16 assists in 39 games. Tibor started the 1982-83 pre-season with Pittsburgh Spirit but was released on September 1, 1982.