Aaron Ramsey equalised with a first-half penalty after Ji Dong-Won had smashed South Korea into the lead.
Moments later Ramsey had the chance to put GB in front with another spot-kick, but it was saved.
The game drifted into extra time and towards penalties. When they arrived, the first nine were scored.
Daniel Sturridge then missed the final one to disappoint the 73,000 fans inside the Millennium Stadium.
Gold medal favourites Brazil await in the semi-finals after they twice came from behind to beat Honduras 3-2 with a penalty from Neymar and two from Leandro Damiao.
Mexico survived a fight-back from Senegal in a thrilling match that delighted a crowd of nearly 82,000 at Wembley who saw Senegal come from 2-0 down to level at 2-2 before two defensive errors granted Mexico to win 4-2 in extra time.
Japan also made it to the last four with a 3-0 win over Egypt that pulled another large crowd of 70,000 at Old Trafford.
Both semi-finals will be played on Tuesday with Mexico facing Japan at Wembley and Brazil playing their game at Old Trafford.
Brazil, chasing their first ever Olympic soccer gold, fell behind when Mario Martinez scored with a stunning curling, dipping volley after 11 minutes, but Honduras were reduced to 10 men when Wilmer Crisanto was sent off for two swift yellow cards in the first half.
Damiao equalised when three Honduras defenders failed to clear in front of goal eight minutes before the break, but the underdogs unexpectedly went back in front after 48 minutes when Roger Espinoza found the net with a soft, low angled shot.
Neymar brought Brazil back into the game with a 50th minute penalty before Damiao made it 3-2 10 minutes later with a well placed shot.
There was even more goals at Wembley were Mexico, 2-0 up midway through the second half, were pegged back to 2-2 by Senegal before Giovani Dos Santos and substitute Hector Herrera scored in extra time to put Mexico into the last four for the first time since they hosted the Olympics in 1968 and lost to Japan in the bronze medal match.
With just over an hour played Mexico appeared to be crusing into the last four, leading 2-0 up after a header from Jose Enriquez and a second from Javier Aquino who pounced on a loose ball in the area and rifled it home.
Senegal, so impressive in the group stage, were never out of contention though and after plenty of chances of their own pulled one back when tournament top scorer Moussa Konate headed home after 69 minutes.
It was his and Senegal’s fifth goal of the tournament, but seven minutes later substitute Ibrahima Balde joined him on the scoresheet when he thundered in an unstoppable header to pull Senegal level.
Japan also made it through to the last four for the first time since 1968 after keeping a fourth consecutive clean sheet with a 3-0 win over Egypt.
The north African side were reduced to 10 men in the 41st minute at Old Trafford in Manchester, where 70,000 fans turned up for the often maligned men’s Olympic soccer tournament.
Japan won with goals from Kensuke Nagai after 14 minutes and late headers from Maya Yoshida and Yuki Otsu.
They took full advantage after defender Saad Saadeldin was sent off in the 41st minute for a professional foul on Manabu Saito.
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