Formentera, who entered the competition for the first time three years ago, sneaked a 1-0 win at 23-times Cup winners Athletic with a headed goal from substitute Alvaro Muniz in the 95th minute.
The goal came from a last ditch corner which goalkeeper Marcos Contreras raced up the pitch to contest.
The Segunda Division B outfit, from an island with a population of 12,000, were heading out of the competition on away goals after a 1-1 draw at their tiny stadium in the first leg but Muniz's goal put them in the last 16 for the first time.
"This is the best day of my career, you cannot understand the happiness I'm feeling right now," said Muniz, whose club almost went bust in 2007.
"I'm not happy for me, this is for all the fans that have travelled, for my team mates, for the players that are injured. Now I would love us to draw Real Madrid or Barcelona."
Coach Tito Garcia had to watch the game from the stands as he was still serving a touchline ban and said he knew his side were in for a great night when Athletic strikers Aritz Aduriz and Inaki Williams missed two late chances.
"That was when I said we would have one last opportunity, and that God was on our side," he said.
Fellow third division outfit Lleida Esportiu also reached the last 16 for the first time after coming from 2-0 down at Real Sociedad to win 3-2 and go through on away goals.
Real Sociedad are one of only nine teams to have won La Liga and have reached the last 32 of this season's Europa League.
Lleida went into the game without a win in their last six games and without scoring in their last five but were full of optimism, blaring out the Spanish pop song "Mi Gran Noche" (My Big Night) on the team bus on their way to the stadium.
"We're going to play it before every game now," said Manu Molina, who scored the equaliser.
The crucial goal was headed in by 17-year-old Bojan Radulovic, who had never scored a goal for the first team.
"When I scored I couldn't believe it, my first goal, at a ground like this, against a Liga side. I didn't know how to celebrate it."
(Reporting by Richard Martin; Editing by Toby Davis)
Share
