With the score locked at 1-1 for the entire second half, neither side could find a decisive goal in regulation time.
South Korea went on to convert four of five shootout attempts while the New Zealanders missed two of four.
The Black Sticks now play Australia in the third-fourth playoff tomorrow morning (NZ time).
They needed to win yesterday's game to earn automatic qualification for the 2016 Rio Olympics, but they had already earned a place at the World League finals in Argentina in December thanks to their top-four placing.
''We had plenty of opportunities to win the game but weren't thorough enough in the circle and attacking third. They defended with lots of numbers and were desperate and we couldn't finish them off,'' Hager said.
''We didn't play as well as we have been and our basics let us down on a couple of occasions.
''It's so disappointing to lose in this way but, at the end of the day, we weren't good enough.''
The Black Sticks opened aggressively and played an up-tempo attacking style that had the Koreans on their heels in the first five minutes.
New Zealand earned a penalty corner midway through the period but Anita Punt's flick was well defended by the Korean runner.
Korea lifted in the second quarter and opened the scoring when Park Mihyun converted a stroke following a penalty corner infringement five minutes out from halftime.
The Black Sticks responded immediately, when Gemma Flynn scrapped for a loose ball in the circle and beat the keeper on the second attempt.
Neither side could find a deciding goal throughout the remainder of the game, and as time wound down, the winner was destined to be found by shootout.
Earlier, the Netherlands progressed to the final with a crushing 5-1 win over Australia, relegating the Hockeyroos to the bronze playoff.
World League
The scores
New Zealand 1
Gemma Flynn
South Korea 1
Park Mihyun
Halftime: 1-1.
Shootout: Korea 4-2.
