Biasion began his motorsport career in 1979 driving an Opel Kadett GT/E in Italian club-level events. He quickly progressed through domestic competition, and in 1983 he won both the Italian and European Rally Championships behind the wheel of a Lancia 037, establishing himself as a major talent in international rallying. Throughout his career he was co-driven almost exclusively by Tiziano Siviero, a partnership that proved exceptionally effective across all surfaces and conditions.
Biasion joined the Lancia works team in the mid-1980s as the squad sought to rebuild following the fatal crash of star driver Henri Toivonen. He became the spearhead of Lancia's Group A programme and, at his peak, was winning nearly three-quarters of the events he entered. His 1988 title made him world champion for the first time; he repeated the feat in 1989, becoming only the third driver to win two WRC crowns, after Juha Kankkunen and Walter Röhrl, and the second to defend his title successfully.
After his championship years, Biasion was unable to recapture that level of dominance. He failed to win a single event for Lancia in 1991, and for 1992 he signed with Ford on a contract that made him the highest-paid rally driver of his era, with some managerial involvement in the programme. His time at Ford proved frustrating. He reportedly described the Ford Sierra RS Cosworth 4x4 as deeply disappointing after his debut with the car at the 1992 Monte Carlo Rally, though he did give it its best WRC result to that point with a second-place finish in Portugal.
Driving the new Ford Escort RS Cosworth in 1993, Biasion won the Acropolis Rally and led the drivers' championship at stages during the season. However, the team's development resources lagged behind rivals, and Biasion was increasingly overshadowed by the quicker François Delecour. When Delecour suffered a road accident that kept him out for much of 1994, the weaknesses in both the car and the team's infrastructure were exposed. Biasion was dropped from the Ford team at the end of that season. He contested a handful of events for private teams before retiring quietly at the close of 1995.
Biasion returned to competitive motorsport through the Dakar Rally and truck racing. He drove for the works Mitsubishi team at the 2003 and 2004 Dakar events. In 2003 he was on course for a podium finish after the penultimate stage, but a gearbox failure at a critical moment resulted in a ten-hour penalty, dropping him outside the top ten overall. In 2004 a roll ended his run despite the team's best efforts at the service park.
He returned to Dakar in 2006 in the truck category with the Iveco works team, partnered by Markku Alén. In 2007 he ran the Fiat Panda 4x4 — nicknamed the "PanDakar" — alongside Bruno Saby as part of a Fiat manufacturer entry. He competed for Iveco again at the 2012 Dakar Rally, winning three stages and finishing sixth overall in the trucks class.
Biasion also claimed the World Truck Championship title in 1998 and 1999 driving an Iveco, demonstrating competitive versatility well beyond his WRC years.
Biasion was married to Italian rally driver Chantal Galli, who won the Italian women's rally championship on multiple occasions. They have four children — Bettina, Isotta, Jacobo, and Olivia. The couple later divorced, and Biasion subsequently married Paola.
Biasion's consecutive WRC titles in 1988 and 1989 represent the high point of Lancia's dominance in the Group A era, and his near-unbeatable consistency during those seasons placed him among the sport's elite. His later career illustrated the difficulty of maintaining front-running form as resources and machinery fell short, but his achievements with Lancia remain a defining chapter in World Rally Championship history.