Jonsson was drafted by the New York Islanders in the 1979 NHL Entry Draft in the 2nd round, the 25th overall pick. During the 1980 World Junior Championships he and Reijo Ruotsalainen were the media's selections as the best defencemen of the tournament. Jonsson made his debut for the Islanders in the 1981–82 season, winning a Stanley Cup championship with the team in his first season. The Islanders repeated as Cup champions again the following season.