Franciszek Gąsior played handball for MKS Pałac Młodzieży Tarnów for nine years from 1961 to 1970. Gąsior then joined Stal Mielec where he won the Polish Cup in 1971 in addition to finishing as the runner-up in 1975. By 1977 he had returned for a second stint at Tarnów, with him staying at the club until 1983, before playing one season in Austria. From 1969 to 1972 Gąsior represented the Polish national handball team, playing in 36 matches and scoring 24 goals. He was part of the team that competed at the 1972 München Olympics, but only played in Poland’s classification match against Iceland, scoring two goals in their 20–17 victory. After his retirement he worked as a teacher and coach.