Simeone, starting in place of injured striker Leonardo Pavoletti, was on hand to get the hosts off the mark after just three minutes when Gianluigi Buffon spilled shots from Luca Rigoni and Lucas Ocampos.
Ten minutes later, Moroccan defender Medhi Benatia allowed Simeone to run in unhindered and direct a diving header past Buffon from Darko Lazovic's pinpoint delivery.
Juve were down 3-0 by the 29th minute when Alex Sandro's attempted clearance from Luca Rigoni's volley saw the ball go over the line and into the roof of the net.
Although Juve grabbed a late consolation through Miralem Pjanic's 82nd minute free-kick, it was too little, too late as the injury-ravaged champions fell to their third defeat of the season.
Now just four points ahead of AC Milan and with a host of fellow challengers close behind, Juventus coach Massimilano Allegri admitted the "blow" of defeat had "brought us back to earth with a bump".
Genoa moved up to ninth and although coach Ivan Juric praised his side's unwavering commitment, he also had the Simeones to thank.
"Before every game I speak to my dad, but not about football -- although it's true he told me I would score against Juve," said Simeone, who was born in Madrid in 1995 while his famous father was playing for Atletico.
Simeone senior was a tenacious, two-way midfielder who won a league and Cup double with Atletico Madrid in 1996, helped Inter Milan to the UEFA Cup (1998) and Lazio to only their second Serie A title (2000).
He also famously lured David Beckham into a sending-off for a petulant, retaliatory kick after fouling the England star during the 1998 World Cup in France.
Giovanni has been watched with interest since signing for Genoa in the summer following a one-season loan deal with Argentine side Banfield.
Although he differs in that he is an out-and-out striker, the confident streak is the same.
"I knew you'd be interviewing me today for something other than my dad," added Simeone, who now has four goals in 10 appearances for Ivan Juric's side.
Juric was keen to play down the individual performances of his players, saying only that Rigoni and Simeone "are doing well".
And amid suggestions Simeone could soon follow injured Juve star Paulo Dybala into the Argentina national side, he is keeping his feet planted firmly on the ground.
"I'm ready for anything," he added. "If there's an opportunity I'll jump at it, but we'll see what comes and when.
"For the moment, I'm just happy to be here and learning from other players like Pavoletti, the coach and from everyone."