Ken Reichel missed a great chance for Eintracht in the first half, shooting high and wide from a promising position, and Vieirinha raced on to a loose ball and blasted it home for the crucial away goal.
Eintracht will stay in the second tier of the Bundesliga where they will be joined next year by relegated Ingolstadt and Darmstadt.
(Reporting by Simon Evans, editing by Ed Osmond)
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