The 50-year-old was sacked as Aston Villa boss last week following a run of five successive Premier League defeats.
Norwich City also dismissed manager Daniel Farke a few hours after securing their first Premier League win of the season against Brentford on November 6.
The Canaries are bottom of the table with five points from eleven games and will host Southampton after the international break.
Craig Shakespeare will join Smith as his assistant manager.
"It has been a whirlwind seven days, but I'm really pleased to be back and working for Norwich City in the Premier League," said Smith.
"Clearly, there has been some wonderful work that has gone into this football club over the last four-and-a-half years. It is now the job of myself and Craig to continue and improve on that work with the ultimate aim of surviving in the Premier League.
"Norwich City is a big club, with a massive hardcore of supporters who are fully understanding in what it means to be part of the club and its community.
"I was brought up in an era when Norwich were competing in Europe - I remember those times well and whenever I've visited Carrow Road and Norwich you can really sense the connection between the fans, staff and players.
"Together, we all have to make Carrow Road a really tough place for visiting teams."
Norwich sporting director Stuart Webber said: "Dean has an outstanding track record and as soon as he made it clear that he was ready to go straight back into a managerial role we contacted him immediately.
"He shares the same values and vision as the club, has a proven track record of winning games in the Premier League as well as developing players and building a winning culture and mentality.
"In Craig Shakespeare we have secured an assistant head coach with an outstanding reputation and vast experience within the game. We are delighted that Craig has agreed to join our new team.
"We look forward to working with Dean and Craig and building on the outstanding work which has been achieved at this club in the last few years.
"There are huge challenges ahead, but we are excited and determined to see how far we can continue to grow as a club on and off the pitch."
