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The win gave Greece a 1-0 aggregate victory after their European playoff first leg ended goalless in Athens four days ago.
Ukraine had their opponents on the back foot for much of the game and missed early chances through Andriy Shevchenko and Olexandr Aliev but it was the visitors who grabbed the winner against the run of play.
Striker Giorgos Samaras sliced open the Ukrainian defence in the 32nd minute with a pin-point pass for Salpigidis who beat keeper Andriy Pyatov from eight metres and sent Greece to the World Cup for only their second time.
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