The Russian beat European silver medallist Zoltan Levai of Hungary in Sunday’s final.
The World Cup is being held in place of the 2020 World Championships, which were due to take place in Belgrade before they called off due to the COVID-19 pandemic.
Elsewhere in the GR finals, Vlasov’s compatriot Emin Sefershaev took the 55kg gold medal with victory over Turkey's Serif Kilic.
The next Russian to taste victory was Nazir Abdullaev, who eliminated Serbian Olympic champ Davor Stefanek before defeating Nanjing 2014 Youth Olympic Games gold medallist Islambek Dadov of Azerbaijan to earn the 67kg gold medal. It was the final bout of Štefanek's career.
The 72kg title went to Hungarian former world champion Bálint Korpási, who was too good for Armenian youngster Malkhas Amoyan in the final.
Russian Olympic champion Davit Chakvetadze was not able to follow his compatriots’ lead in the other finals, going down 10-0 to Belarusian Kiryl Maskevich in the 87kg final.
There could be more joy for Russia on Monday’s Greco-Roman finals, with five wrestlers in action including Rio 2016 bronze medallist Sergey Semenov.