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.
photo Getty Images