The top three teams will qualify for the Paris Games, and Indonesia reached the last four by winning the shootout 11-10 after the score remained tied 2-2 through extra time.
Rafael Struick put Indonesia ahead after 15 minutes only for Komang Teguh's own goal to level the scoreline after 45 minutes. There was still time before the break, however, for Struick to score again.
Jeong Sang-bin equalised with 14 minutes remaining despite Korea being reduced to 10 men minutes earlier when Lee Young-jun was shown a red card.
Lee Kang-hee missed in the shootout, leaving Pratama Arhan to score the winner.
Earlier, Japan knocked out host Qatar 4-2 after extra time to stay on course for an eighth straight Olympic appearance.
Fuki Yamada scored early for Japan but Ahmed Al-Rawi and Jassem Gaber netted to put Qatar ahead.
Seiji Kimura made it 2-2 midway through the second half and after the tie went into extra-time, Mao Hosoya put Japan ahead once more and Kotaro Uchino scored after 113 minutes to secure the win for Japan.
On Friday, defending champion Saudi Arabia face Uzbekistan, while Iraq meet Vietnam.
The team that finishes fourth will face Guinea in a playoff in May with a place in Paris at stake.
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