Dzeko moved on to 55 goals in 97 appearances for the national side with the vital strikes in the Nations League game, showing off his clinical finishing to make the difference in the tie.
As a result, the 32-year-old now sits 10th on the list of all-time leading international goalscorers for a European nation.
Portugal's Cristiano Ronaldo leads the way with 85, while the likes of Ferenc Puskas, Miroslav Klose, David Villa and Gerd Muller also make the list.
Dzeko needs to add another 14 goals to reach the top five.
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Elsewhere, defender Kostas Manolas played all 90 minutes as Greece suffered a 2-0 defeat away to Finland.
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