All Curacao's players successfully scored in the shootouts, while Cong Phuong of Vietnam shot the ball over the bar.
The match's official time ended with a 1-1 draw, a fitting climax for the tournament's final.
At the 57th minute, Jurich Carolina found the net to give Curacao the lead.
Vietnam equalised at the 83rd minute as Quang Hai let the ball roll on to Duc Huy and he calmly tapped it in.
In the latest FIFA rankings, Curacao stand at the 82nd place, while Vietnam are ranked 98th.
Vietnam gave everything they had against a superior team and are able to keep their heads high.
In an earlier match the same day, India earned the third place after beating hosts Thailand 1-0.
On June 5, Vietnam secured its last-gasp victory against Thailand in the opener of the football tournament after a header by forward Anh Duc following a corner kick by midfielder Xuan Truong found the net of its arch rival.
King’s Cup is a friendly four-side competition held annually. This year, Vietnam, India, Curacao and Thailand competed from June 5-8.
It is a friendly tournament but this year it falls during the FIFA international break, meaning it will count towards teams' FIFA rankings – scores that are used to determine seeding at big tournaments.