The 29-year-old qualified for the final with the lowest score but missed only once while his rivals faltered.
He ended up in a shoot-off for gold with world champion Massimo Fabbrizi, and it was the Italian who missed first to give Cernogoraz the title.
Two-time Olympic champion Australia's Michael Diamond led the way in qualifying with a perfect score of 125 but missed five targets in the final - where shooters get one shot instead of two - to blow his chances of gold number three.
Instead, just as he had in Beijing, he lost a shoot-off for bronze, this time to Fehaid Aldeehani of Kuwait.
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