Smith was sacked after just three wins in 13 Championship games.
The assistant manager and first-team coach have left the team.
Allan Russell and Steve Weaver are the caretakers.
"As the club continues to work towards its on-field objectives this season, the board believe that it is the right time to make this transition," Norwich said in a statement.
The club will look to find a new head coach that can deliver both the level of performances required to challenge at the top end of the league whilst implementing a consistent style of play.
Smith questioned the attitude of some fans after the home defeat and said playing away was better for the team.
"I think a lot of them have been waiting for this period to happen or want it to, I don't know," he told the radio station.
It was the third loss in four.
It was a difficult decision, but we felt it was necessary to give ourselves the best chance of promotion this season.
With just under half of the season remaining and three consecutive home games on the horizon, we now have a chance to build some forward and positive momentum, both on and off the pitch.
On 1 October, the Canaries were top of the table, but are now fifth and 12 points behind the leaders.
Smith was appointed as Daniel Farke's successor in November 2021.
During Farke's four and a half years at the helm, he only lasted a single season in the top flight.
After starting his managerial career at Walsall, Smith went on to work at other places.
Villa reached the Carabao Cup final the year after he took them up from the Championship.
After five successive defeats in the premier league, the former defender was sacked, only to be hired by the Canaries eight days later.
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