The 19-year-old Barcelona winger broke the 1-1 deadlock when she latched on to a through ball, cut a Dutch defender inside and out and unleashed a shot that clanged off the post into the net.
The sixth-ranked La Roja, who had not advanced past the round of 16 in two previous two World Cup appearances, will face either Japan or Sweden for a place in the final in Auckland on Tuesday .
All the goals came late in a tight game with Mariona Caldentey scoring Spain's first from a penalty kick in the 81st minute after Dutch defender Stefanie Van der Gragt had handled the ball inside the box.
Van der Gragt, in her final match before retirement, earned some redemption when she equalised in the 91st minute to send the contest to extra time in front of 32,021 fans at Wellington Regional Stadium.
The ninth-ranked Dutch, who were runners-up to the United States four years ago in France, became the latest tournament favourites eliminated in a tournament full of upsets, with the US, Germany, Canada and Brazil already gone.
The Dutch had goalkeeper Daphne Van Domselaar to thank for not falling behind in the 17th minute after she made a fabulous diving save to bat Alba Redondo's close-range header onto the post.
The ball bounced back to Redondo, who pinged a second attempt off the same woodwork.
Spain continued to dominate possession and appeared to have been rewarded with a goal in the 37th minute when Redondo had trouble controlling a cross in front of the net, but Esther Gonzalez poked it in from an offside position.
The Dutch were poised to celebrate a penalty in the 64th minute after forward Lineth Beerensteyn collided with defender Irene Paredes in the area but again VAR intervened.
Beerensteyn had several excellent scoring chances, but the Juventus striker fired one shot wide of the net and another just over the crossbar.
Spain's historic progress comes despite an ugly player revolt in the build-up to the tournament.
Fifteen players declared themselves unavailable for selection in September, saying events with the national team had impacted their emotional and physical health. They directed the bulk of complaints at coach Jorge Vilda.
The Spanish federation backed the coach but only six of the mutineers returned to play at the World Cup.
