Sweden equalised shortly after through Rebecca Blomqvist before Spain captain Olga Carmona struck an 89th-minite winner.
Spain's players were in tears at full-time and celebrated wildly on the pitch after reaching the final for the first time.
"We're just one step away from glory ... We took this little step, and now we need that final push."
Spain had never won a World Cup knockout game before this tournament.
Spain's federation backed Vilda, who froze out the mutineers from his squad before welcoming some of them back for the World Cup.
Persons:
Jorge Vilda, Amanda Perobelli AUCKLAND, Salma Paralluelo, Roja, Paralluelo, Rebecca Blomqvist, Olga Carmona, Vilda, we've, Hritika Sharma, Peter Rutherford
Organizations:
Soccer Football, FIFA, World, Sweden, England, Thomson
Locations:
New Zealand, Spain, Sweden, Eden, Auckland , New Zealand, Australia, Sydney, France, Costa Rica, Zambia, Japan, trouncing Switzerland, Netherlands, Hyderabad