After the 1-1 draw following extra time, Costa Rica keeper Keylor Navas made a superb one-handed stop to keep out Theofanis Gekas’s fourth spot-kick before Michael Umana fired high into the top corner to send the Central Americans into a last eight meeting with Netherlands.
“Last night, I dreamt, this, it seems untrue. I was relaxed because I dreamed it. I dreamed it but I didn’t tell anyone. I felt very confident,” said Umana.
“This is for my family. It’s for the colleagues who got injured before coming here. They’re not with us, but they gave us a hand on the pitch.”
The Ticos, whose only previous appearance in the last 16 came in 1990, will play the Dutch in Salvador next Saturday.
An otherwise dire match was brought to life early in the second half when Bryan Ruiz calmly sidefooted Costa Rica ahead but defender Oscar Duarte was sent off on 66 minutes, allowing the Greeks to find a route back into the match.
With the clock ticking past 90 minutes, Navas could only parry a shot into the path of Sokratis Papasthathopoulos who hammered it into the roof of the net.
After an uneventful first half, the spark the match needed to ignite into a spectacle arrived seven minutes after the restart.
The Greece defence parted to give Ruiz acres of space to sidefoot a Christian Bolanos cross into the net from 15 metres.
Yet their hopes of seeing out the match in routine fashion were scuppered when Duarte received his marching orders after a sliding tackle on Jose Holebas produced a second yellow card and opened the door for a Greek comeback.
Gekas swivelled in the box and his low shot was only parried by Navas into the path of Papasthathopoulos who fired home.
Extra time was largely one-way traffic, with exhausted Costa Rica defenders growing increasingly desperate in their efforts to keep the ball out.
Navas did well to beat away an effort from Lazaros Christodoulopoulos before somehow keeping out a header from Kostas Mitroglou after the clock had ticked past 120 minutes.
Greece coach Fernando Santos was sent to the stands before the shootout started and the tension mounted as seven penalties hit the back of the net before Gekas stepped up to the spot.
The substitute hit it firmly to the keeper’s left but Navas leapt athletically to push it clear with one hand, leaving Umana to thrash his effort into the to left corner.
(Reporting by Toby Davis; editing by Justin Palmer)