Dumitru, set to retire after the Games, had earlier seen off the challenge of two-time world silver medallist Tomoko Fukumi in the semi-final.
However, Menezes gradually got on top of what was a tight encounter, landing a crucial scoring throw with under a minute left and followed with another in the closing stages to secure the title.
Top seed Fukumi left empty-handed after she was beaten on golden score in the bronze-medal fight-off by Eva Csernoviczki of Hungary.
Charline van Snick of Belgium claimed the other third place after overcoming Argentina fighter Paula Pareto, also in sudden death, on golden score.
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