Florida's Nick Cousins took the puck past Swedish Toronto forward Calle Jänkrok before shooting the decisive 3-2 goal, 15 minutes and 32 seconds into extra time.
The game went to extra time after William Nylander equalized to make it 2–2 barely 16 minutes into the third period.
Aaron Ekblad gave Florida a 1–0 lead, played by Gustav Forsling.
With the away win, Florida has advanced 4–1 in the quarterfinal series. In the semifinals, Carolina provided the opposition, who knocked out New Jersey 4–1 in games.
Tre Kronor thus has the opportunity to ask Nylander if he wants and can join the Swedish World Cup squad. The Ice Hockey World Championships, which will be held in Finland and Latvia, began on Friday.
Sweden's head coach Sam Hallam is keeping five spots open on the World Cup squad for potential NHL reinforcements and a dream name has been William Nylander.