Des Fountain suffered a torn ACL in his left knee, sustained in the opening minutes of Aironi's defeat by Ulster at the weekend.
Penney moves to Stadio Zaffanella after four years in London, where the 32-year-old scored 12 tries.
"It's a fantastic chance for me to play in a different country and experience a new environment," said Penney.
"I have enjoyed four amazing seasons at Saracens and felt part of something special.
"I am now grateful to the club for letting me pursue this new and challenging opportunity in Italy."
Director of rugby Brendan Venter added: "Rodd has been a fantastic player for us. He has never been less than committed and has never, ever let the club down.
"We wish him well in Italy, and will obviously keep in touch."
Aironi are in Wales this evening as they face the Scarlets in search of their first league victory.
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