Despite the Swiss being reduced to ten men for the final 13 minutes of the 90 and the whole of extra time, they earned a battling 1-1 draw in St. Petersburg, Russia, but Spain eventually won 3-1 on penalties to progress to the last four.
Schär - who cooly converted his penalty in the round of 16 on Monday night - saw his effort saved by Unai Simón, as did Borussia Dortmund defender Manuel Akanji, while Ruben Vargas blasted over as Switzerland were denied a first ever semi-final appearance at a major tournament.
But Vladimir Petkovic's team had already progressed further in this competition than La Nati ever had before, and they were arguably in the ascendancy at the Gazprom Arena before Remo Freuler was shown a straight red card in the 77th minute.
Liverpool's Xherdan Shaqiri had cancelled out Denis Zakaria's unfortunate eight-minute own goal before Atalanta midfielder Freuler was dismissed by referee Michael Oliver.
Schär came on for Zakaria in the 100th minute, helping his side to force the shoot-out despite their numerical disadvantage. The Newcastle United defender played in four of Switzerland's five games in the tournament.
Switzerland's exit means there are no Magpies players now involved in Euro 2020, although assistant head coach Graeme Jones, who is part of England's coaching staff for the tournament, will be hoping to help Gareth Southgate's Three Lions see off Ukraine and reach the semi-finals on Saturday night.