The England forward finished off a quick counter just after the hour mark.
Erling Braut Haaland thought he had put the visitors ahead before then but his strike was ruled out by the video assistant referee.
The hosts were also denied by VAR after a Marcus Thuram finish was disallowed for offside.
It is the fourth defeat in a row suffered by Gladbach since their coach Marco Rose announced his intention to join opponents Dortmund next season.
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