The Sinkovics, two-time world champions, overhauled Lithuania in the final 500m to win by 1.11 seconds.
Norway's Olaf Tufte, a two-time gold medallist in the single sculls, and Kjetil Borch finished third.
British duo Jonathan Walton and John Collins, making their Olympic debuts, finished almost 11 seconds adrift of the victors in Rio.
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Brothers Martin and Valent Sinkovic won Olympic gold for Croatia in the men's double sculls, with Britain fifth.