The world champions dominated the game at Parc des Princes and had created several chances when Gomez, starting his first match in the tournament, earned the points with a deflected effort from inside the box on the half-hour.
Germany advance from the section with seven points from three games and a better goal difference than Poland, who beat already-eliminated Ukraine 1-0 to qualify in second place.
Northern Ireland, making their first appearance at a European Championship, finished third on three points and may still reach the knockout stage as one the four best third-placed teams although their fate is out of their hands.
(Reporting by Patrick Vignal; Editing by Ken Ferris)
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