With the match tied at 1-1, the Blue Samurais were equal on points with China, who earlier beat Australia 4-3 in a thriller, but Kakitani broke the deadlock in the 91st minute to spark wild celebrations among the Japanese players.
A victory against Japan would have kept Korea in the mix for the title but the partisan crowd at the Jamsil Olympic Stadium were stunned when Kakitani scored in the 25th minute, latching on to a long pass after beating the offside trap.
The lead lasted only eight minutes for the Asian champions, who were the first team to qualify for the 2014 World Cup in Brazil, as Yun Il-lok equalised for South Korea with a strike from outside the box.
Korean keeper Jung Sung-ryong made a valiant save to deny Genki Haraguchi but the ball went to Kakitani who calmly slotted in the rebound in the first minute of additional time to force the result.
"Obviously the result was very important for us," Japan coach Alberto Zaccheroni was quoted as saying by the Kyodo news agency.
"The players gave every last drop of energy they had, especially in the second half. They were fantastic."
Earlier, holders China picked up their only victory of the tournament to stay in the hunt for the title as they staved off Australia for a 4-3 win amid a flurry of late goals.
China, who failed to qualify for the 2014 World Cup, dominated the young Australian side from the outset and took the lead in the fourth minute through Yu Dabao.
In the 29th minute, Aaron Mooy's long-range piledriver from 25 metres out briefly restored parity but China came back stronger in the second half with goals from Sun Ke, Yang Xu and Wu Lei giving them a 4-1 lead.
The Chinese tested debutant goalkeeper Mark Birighitti with a host of long-range efforts but he made some valiant stops to save Australian blushes before Sun doubled their advantage just before the hour mark.
Yang's strike in the 86th minute and Wu's goal a minute later came from defensive lapses with the scorers slotting the ball past the advancing Birighitti after breaking away.
The Socceroos, who finished the event with a draw and two defeats, responded strongly, getting two consolation goals through Adam Taggart and Mitchell Duke.
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- South Korea