Defender Lee pounced at the back post after just seven minutes to convert Ki Sung-Yong's curling free kick before Manchester United winger Park doubled the advantage on 53 minutes to leave Greece with no way back.
It looks like being a short trip to South Africa for Otto Rehhagel's side as Greece's woeful goal drought continues.
This might be their second finals but the Greeks have still to find the back of the net in a World Cup game.
But it could have been so different had Greece, who included four members of their triumphant Euro 2004 winning side, took a chance that fell to them inside three minutes.
Giorgos Karagounis's corner found the unmarked Vasilis Torisidis but he blazed high and wide from close range.
The Koreans took a firm grip on the contest after Jung-Soo fired them ahead and Huh Jung-Moo's men should have been out of sight by half time.
Greece rode their luck after Jung-Soo's opening goal but it finally ran out eight minutes into the second half and Rehhagel's men only had themselves to blame.
Park capitalised on a misplaced pass from Loukas Vyntra to weave beyond two helpless defenders and produce a clinical finish to make it 2-0.
It was no more than the Koreans deserved after dominating Rehhagel's side from start to finish and this win now offers them a realistic chance of reaching the knock-out stages.
It got so desperate for Rehhagel that he even hauled off his captain Giorgos Karagounis at the start of the second half but it made little difference.
The Koreans refused to take their foot off the pedal and could have run up a huge score in the second half but continued to waste more chances as the Greece defence seemed to capitulate.
But this will not have bothered manager Huh Jung-Moo, who will have taken great satisfaction from watching the Koreans stroll to an opening win in Group B.