Sweden allowed Greece to level twice in the absence of star striker Henrik Larsson and Greece were worthy of the draw.
Larsson's wife is expecting a baby and in his absence Greece were able to enjoy significant possession.
They almost took the lead in the 23rd minute but Demis Nikolaidis' shot was disallowed for offside but Sweden responded in the 31st minute when Pontus Farnerud on his debut switched play with a back-heel to find Magnus Svensson for a neat finish.
Defender Sotiris Kirgiakos then had to clear a Markus Allback chip off the goal line after it dropped behind goalkeeper Antonis Nikopolidis.
Panagiotis Fissas headed the equaliser in the 54th minute from close range when he connected with a Nikolas Dabizas header across goal but the home side were again vulnerable on the break.
Substitute Stefan Selakovic was put clear to round Nikolaidis and finish but poor defending prevented Sweden from starting World Cup year with an impressive away win.
An unmarked Giorgos Karagounis was left unmarked in the penalty area to meet a low cross and smack it in.
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