South Korea have won the Asian Cup twice - in 1956 and 1960
South Korea were on the brink of elimination after they fell behind to half-time substitute Abdullah Radif's goal 36 seconds into the second half.
But Cho Gue-sung dramatically equalised in the ninth minute of injury time.
Wolves forward Hwang Hee-chan scored the decisive penalty after two Saudi players had missed.
Jurgen Klinsmann's side will face Australia in the last eight in Qatar on Friday, 2 February.
South Korea are attempting to win the Asian Cup for the first time in 64 years.
-- Courtesy of BBC Sport