After the game finished 1-1, both United players converted their spot-kicks but Chelsea's Daniel Sturridge missed the decisive ninth penalty as Korea progressed to a semi-final against Rafael's Brazil.
Rafael earlier played the full 90 minutes as his country twice came from behind to beat Honduras 3-2 at St James' Park and book their place in the last four.
The Reds full-back will play at Old Trafford on Tuesday (19:45 BST) as Brazil take on Korea for a place in the gold medal final against either Mexico or Japan.
Korea took a deserved lead in the 29th minute against GB when Sunderland striker Ji Dong-won's swerving shot from range deceived keeper Jack Butland.
Arsenal's Aaron Ramsey equalised from the penalty spot seven minutes later following a handball, but then missed a second spot-kick before the break.
Cleverley once again looked sharp in midfield and played the entire game, while Giggs assumed the captain's armband when he replaced Craig Bellamy in the 85th minute, but couldn't conjure up an extra-time winner for Stuart Pearce's side.