After a goalless draw over 120 minutes of the last 16 clash in Frankfurt, Porto goalkeeper Costa produced three saves to thwart Slovenia's three takers as his team claimed a 3-0 shoot-out win.
Cristiano Ronaldo, who missed a penalty in extra-time, converted in the shoot-out along with Bruno Fernandes and Bernardo Silva for Portugal, who face France on Friday in the last eight.
The 24-year-old also made a vital one-on-one save with his foot in the second period of extra-time when Slovenia's Benjamin Sesko bore down on his goal following a defensive error.
"I think this is probably the best game of my life, maybe the game I was most useful in," Costa told reporters.
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Kriel's Springbok dream is still alive
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