Abdullah Radif's second-half strike looked for much of the match like being the winner for Saudi Arabia, but Cho Gue-sung dramatically headed home in the 99th minute to equalise and send the match to extra-time.
The scoreboard was not troubled again as a heavyweight last-16 clash went to the spot, where Sami Al Najei and Abdulrahman Ghareeb both had their penalties saved. South Korea had no such issues, converting their first three to set the stage for Hwang to seal a 4-2 shootout win and book a quarter-final date against Australia.
Jurgen Klinsmann's side were not at their best in the group stage, finishing as runners-up to Bahrain, and it left them facing a tricky clash against Saudi Arabia.
The first-half was a relatively uneventful affair, until Saudi Arabia twice hit the bar in a matter of seconds before the break. Saleh Al Shehri thundered an effort against the woodwork from a corner, and Ali Lajami then looped a header of his own onto the bar from the rebound.
Roberto Mancini introduced Radif at half-time and it proved an inspired decision, as the forward opened the scoring 32 seconds after the restart, finishing brilliantly after Salem Al Dawsari's flick played him through on goal.
South Korea swiftly brought on Hwang as they pushed for an equaliser, and the pressure was piled on the Saudi Arabian defence in the closing stages.
Ahmed Ali Al Kassar was forced into two brilliant saves late on, while Hassan Tambakti twice scrambled the ball away off his own line, the second of those coming as Heung-min Son fired in an effort from the edge of the box.
Ten minutes of stoppage-time gave South Korea hope and Cho should really have forced extra time, as the ball was stood up to the back post and he somehow headed against the bar from only a few yards out.
Hwang then dragged an effort inches wide of the far post when clean through, as an increasingly panicked Saudi Arabia was exposed, before a deserved equaliser eventually came. The ball was whipped to the back post, nodded back across goal by Young-Woo Seol and Cho was on hand to provide the finishing touch.
South Korea looked the more likely to edge in front in extra time, though wasted their big chance as Son's strike was blocked by team-mate Cho. Radif was denied in the dying seconds, as Jo's save took it to a penalty shootout.
It was a sign of things to come, as he produced two brilliant saves before Hwang had the final say.