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Portugal struggled to create much aside from set pieces in the first half, with Miguel Veloso firing an effort wide from the edge of the box when presented with a rare sight of goal.
Greece were also short of quality in the final third, coming closest in the 38th minute when Vassilis Torosidis curled an effort well off target from 25 yards.
The hosts began to look more threatening after the restart but still couldn't find a breakthrough despite Nani fizzing a dangerous low ball across the face on goal in the 54th minute.
Nani was consistently Portugal's biggest threat and came close again with 20 minutes remaining as his strike was eventually gathered by Karnezis at the second attempt.
Portugal then almost grabbed a dramatic late win when Costa's cross took a rebound that diverted it against the woodwork with virtually the last kick of the match.