Spain will now face France or Germany for a place in the final.
For the best part of an hour, the hosts defended doggedly and attacked on the break, but once Spain found their groove they were able to quickly kill the game off with two expertly taken goals.
Spain looked to have made the perfect start when Mariona Caldentey was upended by a rash challenge by Nadine Riesen, but having opted to take the spot-kick herself the forward rolled it wide of the post.
Pina had several chances for Spain but stout defending by the hosts saw them go in level at the break, much to the delight of the raucous crowd.
It took a moment of brilliance from Aitana Bonmati to break the deadlock for the world champions in the 66th minute, teeing up Athenea with an instinctive backheel for the substitute to guide the ball past Swiss keeper Livia Peng.
Five minutes later, Pina doubled her side's advantage as Switzerland captain Lia Waelti was dispossessed just outside her own penalty area, and Pina curled a superb shot into the top corner before wheeling away in celebration.
The drama continued right to the end with Spain's Alexia Putellas missing a late penalty and Switzerland's Noelle Maritz shown a straight red card for a stoppage-time challenge.
As the Spaniards danced in a circle to celebrate advancing to the last four, the Swiss fans sang loudly in appreciation of the team, and the players bounced together in unison as the hosts exited the tournament with their heads held high having made the knockout stages for the first time.
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