The 50-year-old Smith, who was axed by Villa on November 7 after five successive defeats, takes over at Carrow Road on a two-and-a-half-year contract, the club said.
Smith succeeds Daniel Farke, who was sacked on November 6 with Norwich rooted to the bottom of the Premier League.
"It has been a whirlwind seven days, but Im really pleased to be back and working for Norwich City in the Premier League," said Smith, who will be assisted in the dug-out by assistant head coach Craig Shakespeare.
"Clearly, there has been some wonderful work that has gone into this football club over 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."
Former Chelsea manager Frank Lampard was believed to have been Norwich's initial top candidate, but the ex-England star pulled out of the running once Smith emerged as a rival for the job.
Farke's spell as Norwich boss ended on the day he led them to their first league win this season, a 2-1 success against Brentford.
The German had overseen Norwich's promotion from the Championship last season, but struggled to lift the club away from the Premier League relegation zone despite a host of new signings.
Norwich's next Premier League game is at home to Southampton on November 20 and Smith said he could not wait to get going.
"From the age of 16 I've been working in football. I think in that time I've had four months out and didnt enjoy it," he said.
"It's great to get straight back in with a club that are determined to be progressive."
Standout matches today