Greece, who joined Bosnia on 16 points from seven games in Group G, produced a dismal first half and were really struggling when Jose Holebas was given a second yellow card for tangling with an opponent and sent off in the 59th minute.
The numbers were evened seven minutes later when the hosts had David Hasler sent off, also for a second bookable offence.
Greece eventually broke through in the 72nd minute when Mitroglou turned in a low cross from Vassilis Torosidis from six metres. Sotiris Ninis hit the post from distance as Greece dominated the closing stages.
A frustrating night for Greece turned to joy when they learned rivals Bosnia had suffered a surprise 1-0 home defeat by Slovakia.