Reiner's own goal 10 minutes from time proved the difference as the visitors climbed to seventh.
Neither side entered the fixture in good form, with Academica having won just one match this season, and the first half was goalless.
Both teams chased the result with attacking substitutions but it was Rio Ave who snared the winner late on.
Brazilian Reiner was the scorer, with the Academica defender putting through his own net.
Academica sit 14th in the table as a result of their latest defeat.
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