Wassberg's skiing idols when growing up were Sixten Jernberg and Oddvar Brå. He has described his mental strength and physical fitness as his greatest abilities as a skier, with his main weakness being a lack of sprinting ability.
He won the World Cup in 1977. In 1980 he received Bragdmedaljen, an award given to the best Swedish sportsman/sportswoman each year. He did however for many years refuse to accept the award in protest to an earlier decision by the award committee, but finally in 2013 he accepted the medal.
Wassberg won four gold Medals at the Winter Olympics: in 15 km (1980), 50 km (1984), and the 4 x 10 km relay (1984, 1988). At the FIS Nordic World Ski Championships, he earned three golds (50 km: 1982, 30 km: 1987, and 4 x 10 km: 1987), three silvers (15 km: 1985, 1987; 50 km (1987), and one bronze (4 x 10 km: 1985). Additionally, Wassberg won the 50 km at the Holmenkollen ski festival three times (1980, 1982 and 1987) and the 15 km twice (1979, 1985).
At the 1980 Winter Olympics in Lake Placid, Wassberg edged out Finland's Juha Mieto by 0.01 seconds in the 15 km, the closest cross-country skiing race in Olympic history. Wassberg subsequently suggested to Mieto that the gold medal should be split between them "as one one-hundredth of a second is nothing in a 15-kilometer race". This incident led the FIS to change their timing to the nearest one-tenth of a second. It also resulted in an apocryphal urban legend that Wassberg and Mieto's medals were cut in half and re-welded into half-gold, half-silver medals. At the 1984 Winter Olympics, Wassberg beat out fellow Swede Gunde Svan by 4.9 seconds in the 50 km, the closest margin of victory ever in that event until Giorgio Di Centa (Italy) edged out Yevgeny Dementyev (Russia) by 0.8 seconds at the 2006 Winter Olympics though the 2006 event was a mass start event while the 1984 event was an interval start event.