Competing in the last eight for the first time having disposed of Villarreal and Racing Santander, Mirandes shocked the crowd at a chilly Cornella stadium in Barcelona by opening a 2-0 lead thanks to Alain Arroyo and captain Pablo Infante.
Espanyol, who have won the trophy four times but not since 2006, looked well beaten ahead of next week's second leg until Vladimir Weiss stole in to make it 2-1 five minutes from time.
The Segunda B leaders, based in the town of Miranda de Ebro in northern Spain, have an annual budget of 1.23 million euros compared with Espanyol's estimated 50 million.
"Complaining about the referee is a very poor argument," he told a news conference. "If he didn't give a penalty it wasn't one."
The winners of the tie will play Athletic Bilbao or Real Mallorca in the semi-finals. The top-flight rivals meet at Bilbao's San Mames stadium on Wednesday.
Holders Real Madrid host arch rivals Barcelona, the team they beat 1-0 in last year's final, later on Wednesday and Levante visit city neighbours Valencia on Thursday.
