The former Barcelona full-back was carried off on a stretcher with 14 minutes left to play at the Stadio Luigi Ferraris, after hurting himself following a clumsy challenge on Lucas Ocampos.
Tests have now revealed that he has sustained a compound fracture and is therefore likely to face a lengthy spell on the sidelines, with further examinations set to take place on Monday.
"The first assessments carried in hospital have revealed a compound fracture of the fibula of his left leg, which will be re-evaluated tomorrow [Monday]," a Juventus statement confirmed.
Leonardo Bonucci is also set for tests back in Turin after he picked up a thigh muscle injury in the shock defeat.
"With regard to Leonardo Bonucci, a strain to the left thigh was detected, which will also be reassessed," the club said.
Two goals from Giovanni Simeone - the son of Atletico Madrid head coach Diego - helped Genoa to a win that means Juve's lead at the top of Serie A remains at four points.