Mattiacci was appointed as Stefano Domenicali's successor in April, but failed to turn the Marinello outfit's season around - Ferrari finishing fourth in the constructors' championship, while Fernando Alonso and Kimi Raikkonen were sixth and 12th respectively in the drivers' standings.
Arrivabene arrives at Ferrari having held a number of roles with tobacco company Philip Morris, which maintains close ties with Ferrari through its Marlboro brand.
Indeed, Arrivabene - a board member at Juventus football club - has represented sponsors on the Formula One Commission since 2010.
"We decided to appoint Maurizio Arrivabene because, at this historic moment in time for the Scuderia and for Formula One, we need a person with a thorough understanding not just of Ferrari but also of the governance mechanisms and requirements of the sport," said Ferrari chairman Sergio Marchionne.
"Maurizio has a unique wealth of knowledge: he has been extremely close to the Scuderia for years and, as a member of the F1 Commission, is also keenly aware of the challenges we are facing.
"He has been a constant source of innovative ideas focused on revitalisation of Formula One. His managerial experience on a highly complex and closely regulated market is also of great importance, and will help him manage and motivate the team.
"I am delighted to have been able to secure his leadership for our racing activities."
Marchionne also paid tribute to Mattiacci following his brief spell at the helm.
He added: "We would also like to thank Marco Mattiacci for his service to Ferrari in the last 15 years and we wish him well in his future endeavours."