Mattiacci lasted only seven months on the job after being hired in April to replace Stefano Domenicali, who resigned.
On Sunday, Ferrari finished its first season without a race win since 1993.
Last week, the Italian automaker announced that Sebastian Vettel was replacing Fernando Alonso as the team's leading driver alongside Kimi Raikkonen.
Arrivabene comes to Ferrari from Philip Morris, which he joined in 1997 after a 20-year career in marketing and promotions in Italy and abroad. He was most recently vice-president of consumer channel strategy and event marketing with Philip Morris, working closely on the partnership with Ferrari.
"We need a person with a thorough understanding not just of Ferrari but also of the governance mechanisms and requirements of the sport," said Fiat-Chrysler CEO Sergio Marchionne, who took over as Ferrari chairman from Luca Di Montezemolo in September.
Since Alonso's triumph at the Spanish Grand Prix in May 2013, Ferrari has gone 33 races without a win. And while Vettel won 13 races and his fourth consecutive world title at Red Bull last year, this season he managed only four podiums and finished fifth in the standings, 217 points behind F1 champion Lewis Hamilton.
"I know that there is a mountain of work waiting for me, and that is part of the task that I have chosen," Vettel said after Sunday's Abu Dhabi Grand Prix. "I think if you climb a mountain together, that is a very strong bond, and then ending up successful, that is a fantastic outlook."
Alonso finished sixth in the drivers' standings, Raikkonen placed 12th and Ferrari ended fourth in the constructors' rankings, a whopping 485 points behind champion Mercedes.
Red Bull and Williams also finished ahead of Ferrari, and things could get even tougher next season with Honda returning to the sport at McLaren.
Arrivabene has also represented all Formula One sponsors on the F1 commission since 2010. And he's been an independent member of the Juventus football club's board since 2012.
"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," Marchionne added. "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."
Mattiacci was previously the president of Ferrari North America.
"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," Marchionne said.