The Sounders emerged from the jaws of defeat in an all-time classic, having trailed Minnesota 2-1 approaching the 89th minute in the Western Conference final.
Raul Ruidiaz equalised in the final minute of regulation before substitute Svensson completed the epic comeback in the third minute of stoppage time in Seattle, where Minnesota had led 2-0 after 67 minutes.
The Sounders, who will meet Columbus Crew in their fourth final in five years, became the first team since 1999 to come from two goals down to win a play-off match in 90 minutes.
Minnesota looked on track to advance to its first final thanks to Emanuel Reynoso and Bakaye Dibassy.
Reynoso curled a free-kick past Sounders goalkeeper Stefan Frei in the 29th minute – the Minnesota star's opener the club's first direct set-piece goal.
Dibassy doubled the lead with 23 minutes remaining as Reynoso turned provider to become the fourth player in MLS history to be involved in multiple goals in three consecutive play-off fixtures, following in the footsteps of Jozy Altidore, Guillermo Barros Schelotto and Landon Donovan.
Will Bruin came off the bench to trigger Seattle's comeback in the 75th minute after the Sounders forward scored within two minutes of his introduction.
Ruidiaz shocked Minnesota when he riffled a shot into the net at the death, before Svensson headed home the stoppage-time winner – there were four minutes, one second between the equaliser and winning goal.
Two different substitutes scored in the same match for the Sounders for the second time in Seattle's MLS history, after Obafemi Martins and Leo Gonzalez in 2015.