Born in Goteborg, Sweden, Johansson played with the Vastra Frolunda and Bjorkloven domestic teams. In 1985 he represented the host country at the European Junior Championships and was selected to the tournament all-star team. He also played at the 1986 and 1987 World Junior Championships and was selected the best defenceman at the latter.
Johansson was the first round draft pick, 14th overall, by the Buffalo Sabres at the 1985 NHL Entry Draft. He joined Buffalo in 1987-88 and impressed with 42 points and selection to the NHL All-Rookie Team. That year he led all first year players with a plus/minus rating of +12 and finished second with 38 assists. He started slowly the next season and was included in a multi-player deal with Washington that brought goalie Clint Malarchuk to the Sabres.
Upon his trade to Washington in 1988-89, Johannson became a fixture on the team's blueline. He scored at least 40-points four times and was a major part of the Caps' specialty teams and transition game. The talented blueliner helped Washington reach the semi-finals in 1990 and the final in 1998. In 1999-00 the veteran rearguard led all NHL defenceman with 2,389 shifts.
Johansson's went on to play 15 seasons in Washington before opting to call it a career at the end of the 2002-03 season. However, the Toronto Maple Leafs signed the veteran rearguard at the 2004 trade deadline for the remainder of the 2003-04 season, in hopes of adding some depth to their blueline. The Leafs failed to win the cup that season and Johansson retired for good after the playoffs.
Internationally, Johnasson played for Sweden when they won consecutive World Championships in 1991 and 1992 and claimed silver in 1997. He was also a member of his country's team at the 1991 Canada Cup, the 1996 World Cup of Hockey and the 1998 Nagano Olympics.