Sturridge saw his spot-kick saved by substitute goalkeeper Lee Bum-Young after all eight of the previous penalties had been converted, and Celtic's Ki Sung-Yeung netted to send his side into the last four.
Aaron Ramsey had earlier cancelled out Ji Dong-Won's opener from 12 yards and then missed another spot-kick as the two sides played out a contest low on quality but high on drama.
The result ends Team GB's involvement in the Olympic football after the women's side crashed out to Canada at the same stage on Friday evening, with the nature of this defeat all the more galling.
Stuart Pearce stuck with the same starting XI which saw off Uruguay to progress as Group A winners on Wednesday, with Ryan Giggs once again having to be content with a place on the bench. South Korea, meanwhile, made two changes to the team which drew with Gabon, with Nam Tae-Hee and Sunderland's Ji Dong-Won recalled.
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South Korea hassled and harried Team GB, pinning their rattled opponents back and tempting them into conceding corners and committing fouls deep inside their own half. The consistently dangerous deliveries from Celtic’s Ki came close to making them pay a number of times, and on 18 minutes Arsenal's Park Chu-Young headed over unmarked.
A breakthrough looked simply a matter of time, and so it proved on 29 minutes when Ki's lay-off found Ji around 20 yards out, and the Sunderland man's swerving strike deceived Butland as it flew into the far corner.
Stung into action, Team GB poured forward with greater purpose, and were granted a chance to level the scores when Ryan Bertrand's scuffed shot struck the arm of a sliding Oh Jae-Suk in the penalty area. The right-back was booked for his trouble, and Ramsey's low spot-kick just about crept under Jung Sung-Ryong.
All of a sudden, panic reigned in the South Korea defence and five minutes later Sturridge went down under the clumsy challenge of Hwang Seok-Ho.
Once again referee Wilmar Roldan pointed to the spot, and once again Ramsey stepped up. His second effort was no more tame than his first, but this time Jung parried it away and the Welshman was left cursing a golden opportunity missed as the half-time whistle sounded.
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South Korea found themselves unable to keep up their high-intensity pressing game as fatigue levels gradually decreased, giving Great Britain more time to settle into their passing game and look to craft attacks. Kim Young-Gwon, however, highlighted the threat they still posed on 63 minutes with a fierce free kick which flew just over.
Team GB's likeliest route to goal looked to be with Bellamy down the right flank and, two minutes later, a flowing one-touch passing move involving the Liverpool man and Ramsey set Sturridge racing through on goal, only for Ki to make a superb covering tackle.
As the closing stages approached chances grew fewer and further between. Ji caused panic in the Team GB defence on 84 minutes when he headed Ki's floated ball in dangerously back across goal, but none of his team-mates had gambled on a late run into the box.
With five minutes of normal time to go Pearce replaced the tiring Bellamy with Giggs, much to the delight of the frustrated Millennium Stadium crowd, but the Manchester United veteran proved unable to create a decisive breakthrough and the match went into extra time.
It was South Korea who began brighter after the restart, and Butland atoned for his earlier mistake by pulling off an excellent save to deny Koo Ja-Cheol from the angle. Ji pounced on the goalkeeper's parry, but could only direct his header well wide.
Three minutes before the halfway mark the Sunderland striker missed an even more glorious headed chance. Another superb Ki corner found him unmarked around the penalty spot, but he got under the ball and skied his attempt high above a grateful Butland's crossbar.
In the dying moments the tireless work-rate of Sturridge looked to be Team GB's best hope of a winner but, with both sides too weary to carve each other open and too afraid to take the risk, the contest slid intevitably towards penalties.
In the shootout, Ramsey, Cleverley, substitute Craig Dawson and Giggs all netted for Team GB as all of the first eight penalties were converted. But Sturridge saw his effort saved by substitute goalkeeper Lee Bum-Young, and Ki fired high into the net to send South Korea into the semi-finals.