Mitroglou, who scored twice in Greece's 3-1 first-leg win on Friday, beat the offside trap midway through the first half and gave keeper Ciprian Tatarusanu no chance with his 22nd goal in all competitions this season.
Romania equalised 10 minutes after the break when Vasilis Torosidis struck the ball past his own goalkeeper, Orestis Karnezis, from the edge of the area for an astonishing own goal.
Greece, the 2004 European champions, competed in the previous World Cup finals, in South Africa in 2010, as well as the 1994 tournament in the United States, but they have failed to get past the first round, losing five of their six matches.
(Writing by Angel Krasimirov, editing by Stephen Wood)
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