Juve were wretched in the first half and trailed 3-0 inside 29 minutes.
Simeone, son of Atletico Madrid boss Diego, turned in a third-minute opener after Gianluigi Buffon's triple save and headed home his second. Alex Sandro's own goal sealed the game.
Juve ended with 10 men because of an injury to Dani Alves, before Miralem Pjanic struck a consolation free-kick.
Alves, who was later diagnosed with a broken leg, was forced off after fellow defender Leonardo Bonucci suffered a game-ending injury in the first half, making a bad day even worse for the Italian champions.
They were awful at the back in the first half and could have trailed by more than three goals.
They were marginally better after the break, bringing £75m striker Gonzalo Higuain off the bench, but never looked like mounting a comeback despite Pjanic's late goal.
Juve, chasing a sixth consecutive title, have now lost three Serie A games, as many as they did in 2014-15 and more than in their successful campaigns of 2011-12 and 2013-14.
They are four points above second-placed AC Milan.
It was the first time Juventus had conceded three goals in the first half of a Serie A game since 2004, when they lost 4-3 to Lecce.
Simeone, 21, doubled his Serie A goals tally in the opening quarter of an hour.
After arriving in the summer from River Plate, the Madrid-born Argentina Under-20 international had scored twice in his first 10 Serie A games.
And he was the star of Genoa's famous win, scoring from six yards out after Buffon had parried his first effort and then heading their second.
He briefly thought he had a 30-minute hat-trick when he lashed home after Sandro cleared the ball, but the ball had narrowly crossed the line following Luca Rigoni's shot.
Genoa had fewer chances in the second half, but Buffon did deny Simeone a third with a good save.
Giovanni Simeone scored two early goals as mid-table Genoa stunned Serie A leaders Juventus.