In the other Pool D match at the last major men's tournament before the London Olympics, Germany defeated Great Britain 1-0 to claim fifth place and ensure qualification for next year's event in Argentina.
Pakistan trounced Korea 6-2 in the pool stages but faced a sterner test on Sunday, with both sides desperate to avoid finishing last in the eight-nation tournament.
In a see-sawing match, Korea took an early lead through Seo Jong-Ho after five minutes but Rizwan hit back immediately.
A defensive lapse allowed Jang Kyu-Yeob to restore Korea's advantage, only for Pakistan captain Muhammad Imran's penalty corner conversion to again cancel it out as the Green Shirts appeared to have finally taken control of the match.
Abdul Khan set up a Muhammad Tousiq field goal with a probing cross just before the break then scored himself shortly after play resumed, dispossessing a Korean defender in the goalmouth and nudging the ball home.
With the luxury of a two-goal buffer, the Green Shirts allowed themselves to relax in the final stages, paying the price when You Hyo-Sik scored in the 54th and 62nd minutes to send the match into extra time.
Pakistan restored some urgency to their game to dominate the extra period, with Rizwan making no mistake from close range when a Shakeel Abbasi pass gave him the opportunity to score the golden goal.
Pakistan team manager Khawaja Junaid said the Green Shirts had benefited from exposure to top teams in their first Champions Trophy appearance since 2007 but needed to improve their consistency.
"We played well for 20 minutes against Australia and for 30 minutes against Germany, now we need to stretch that out for the full 70 minutes," he said.
"That's our major problem and we'll be working very hard on that before the Olympics."
A 30th minute penalty corner conversion from Germany's Thilo Stralkowski was enough for the reigning Olympic and European Champions to edge Great Britain for fifth place.
The British had their chances but failed to score from four penalty corners.
Defending champions Australia will play Spain in the final later Sunday, while New Zealand face the Netherlands to decide third place.
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