Inzaghi replaces Paulo Sousa, who was sacked earlier this week.
Salernitana have not won any of their eight league games this term, picking up just three points to sit 19th on the table.
Their next game after the October international break will be against basement club Cagliari.
Inzaghi is back in management for the first time since leaving Reggina this summer.
The former AC Milan and Juventus striker won the FIFA World Cup with Italy in 2006.
"Salernitana announces that it has reached an agreement with Mr. Filippo Inzaghi and to entrust him shortly with the technical guidance of the first team," the club said.
Speaking at his unveiling, Inzaghi said: “I thank the President, the club and all those who called. I wanted a rest, but I was already fed up and couldn’t take it anymore.”
“I was waiting for the right call to come along. I didn’t know if this football world deserved my honesty anymore, I wanted to make sure my staff and I could get back to working the way we love.
“I wish Sousa all the best, because in my view he did well, and I will try to save this team from relegation because I think we can do it. We need all of you.
“We must bring enthusiasm back to the side and the only promise I can make is that if my team does not battle on the pitch, then I will go home. A team with 10 internationals in the squad cannot be forced to fight against relegation.”
