The Frenchman has replaced Alberto Gilardino in the Serie A side's dugout.
Genoa are currently 17th in Serie A after winning just two of their first 12 matches this season.
Vieira's opening match in charge will come at home to Cagliari on Sunday.
It is his first job since leaving Strasbourg in July following just a single season in charge.
Meanwhile, he will reunite with a former teammate who he already has experience managing.
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Genoa's squad includes former Italy striker Mario Balotelli, who played alongside Vieira 44 times for Manchester City and Inter Milan.
The 34-year-old joined the club last month, having been a free agent after leaving Adana Demirspor in the summer.
He has yet to score for his new side but has picked up a yellow card in each of his first two appearances.
It will be the second time Balotelli has played under Vieira after making 10 appearances for his Nice side in 2018.
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The forward failed to score in those matches, though, and reportedly turned up late for Vieira's first training session.
Addressing the challenge of managing Balotelli at the time, Vieira said: "When it comes to Mario, I want to answer back, or just slam him up against the wall, or leave him hanging by his collar on the coat rack.
"But I can't, as I'm no longer a player."
He reflected on their time working together again in 2020, adding: "Mario's mindset was difficult for a collective sport.
"The philosophy I wanted to put in place, the togetherness and work ethic I wanted to build, it was difficult for me to work with a player like Mario.
"It was really difficult for both of us to work together so we decided to go different ways."