Gold medal favourites Brazil twice came from behind to beat underdogs Honduras 3-2 to reach the semis.
Mexico will face Japan in the other semi-final after they beat Senegal 4-2 after extra time and Japan beat Egypt 3-0.
South Korea will play Brazil at Old Trafford on Tuesday while Japan will play Mexico at Wembley on the same night.
After one of the greatest days in their Olympic history with six gold medals, Britain's football players were looking to make history of their own by reaching the Olympic semis for the first time since 1948.
The match against South Korea ended 1-1 after Aaron Ramsey scored one penalty after 36 minutes and had one saved three minutes later after Ji Dong-Won had put Korea ahead in the 29th minute. There were no further goals in normal or extra time so the game went to a shootout.
After both teams converted their first four penalties, Daniel Sturridge's poor effort was saved by Korean keeper Lee Bum-Young and Ki Sung-Yeung converted to give his side a 5-4 shoot-out win.
Brazil, chasing their first ever Olympic football gold, fell behind to Honduras when Mario Martinez scored with a stunning 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.
Leandro Damiao equalised 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 Leandro Damiao made it 3-2 10 minutes later with a well placed shot.
Brazil coach Mano Menezes said: "The game was very difficult as I expected it to be. If we had scored one or two chances early in the game it might have been different, but the first Honduras goal unsettled the team."
Mexico survived a fightback from Senegal in a thrilling match that delighted a crowd of almost 82,000 at Wembley who saw Senegal come from 2-0 down to level at 2-2 before two defensive errors allowed Mexico to win 4-2 in extra time.
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.
Senegal 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 football tournament.
Japan won with goals from Kensuke Nagai after 14 minutes and late headers from Maya Yoshida and Yuki Otsu.
- South Korea