Video report by ITV News Correspondent Richard Pallot, in Milan
Cases in the country doubled within a few days to more than 200 - the most in Europe.
In the regions of Lombardy and Veneto, a lock-down is in place in several towns, affecting about 50,000 people.
There is concern that unless the outbreak of the virus is contained within Italy, there could be more rapid spread in other European countries.
Civil protection officials said two elderly men in northern Lombardy are among the dead.
Road blocks were set up in at least some of ten towns in Lombardy at the epicentre of the outbreak, including in Casalpusterlengo, to keep people from leaving or arriving.
Buses, trains and other forms of public transport — including boats in Venice — were being disinfected, Mr Zaia told reporters.
Museums were also ordered to shut down after Sunday in Venice, a top tourist draw anytime of the year, as well as in neighbouring Lombardy.
Shannon Belvin, a 22-year-old British tourist in Venice, said "literally everything is closed."
"It does seem like people are on edge. I was on a tour and half the boat was wearing masks. There are quite big groups of tourists from east Asia that are all wearing masks," she said.
Other northern regions with smaller numbers of cases are Emilia-Romagna and Piedmont.
ITV News Correspondent Richard Pallot reports from Italy
Italy has also tested millions of airport passengers arriving from other places for any signs of fever.
“Worry is understandable, panic, no,” Premier Giuseppe Conte told a state TV interviewer.
Mr Gallera told reporters in Milan that schools, museums, discos, pubs and cinemas would stay closed for at least seven days.
But restaurants in Milan and other Lombardy cities outside the main cluster area can still operate since, unlike at concerts and other entertainment venues, in eateries “people are not congregated in one place and there is space between tables,” Mr Gallera said.
Lombardy’s ban on public events also extended to Masses. Venice also was forbidding public Masses, while in Milan, the city’s iconic Gothic cathedral was closed to visitors. School trips inside Italy and overseas were banned.
But in the south, thousands turned out for a visit by Pope Francis in the port city of Bari. The pontiff shook hands with many of the faithful.
In Lombardy, a populous region which includes the country’s financial capital, Milan, nearly all the cases of Covid-19 were in the countryside, mainly in Codogno and nine neighbouring towns.
In those towns, only grocery stores and pharmacies were permitted to open, and people were not supposed to enter or leave the towns.
Melissa Catanacci, who lives on one of Codogno’s main roads, said in the morning, she ventured outside for a stroll along with her husband and two children, ages 10 and 13.
“Every quarter-hour or so a car goes by” on the main road, Ms Catanacci said, speaking by telephone.
With businesses closed, the usual Sunday “passeggiata” — a leisurely stroll through local streets — did not last very long, she said.
Sporting events were cancelled, from children’s team practices to Serie A football matches which were to be played in northern stadiums. Those measures were ordered on Saturday night by the Italian government.
Dispensers of hand disinfectant were being installed in trains run by the state railways, which also said it was supplying its crews with masks and disposable gloves.
Italians travelling abroad were already feeling the effects of a crackdown, with a bus from Milan barricaded by police in the French city of Lyon for health checks and an arriving Alitalia plane blocked on the tarmac in the African island nation of Mauritius.
Concern was also on the rise in neighbouring Austria, which halted all rail traffic to and from Italy for several hours after suspicion that a train at its southern border with Italy had two passengers possibly infected with the virus on board, authorities said.
Austria’s interior ministry said it had been informed by Italy’s railway company that two passengers had a fever and stopped the train at the Brenner crossing before it could enter Austria.
However, just before midnight Austria’s Federal Railways announced on Twitter the ban had been lifted.
Austrian Interior Minister Karl Nehammer said the two people suspected of being infected with the virus on the Eurocity 86 train from Venice to Munich had tested negative and the train would be allowed to continue on its way, according to the ORF broadcast network.
A decision to call off Venice Carnival was announced on Sunday by Veneto regional governor Luca Zaia as the number of confirmed virus cases soared to the largest number outside Asia.
“The ordinance is immediately operative and will go into effect at midnight,” said Mr Zaia, whose area includes Venice, where thousands packed St Mark’s Square.
Top-flight football matches were also postponed.
Italy’s first two cases were a Chinese tourist couple, diagnosed earlier this month and reported recovering in a Rome hospital.
The death on Sunday of an elderly woman, who was already suffering from cancer when she contracted the virus, raised the nation’s death toll to three, said Lombardy regional official Giulio Gallera.
Authorities expressed frustration that they have not been able to track down the source of the virus that is spreading in the north and which surfaced last week when an Italian man in his late 30s in Codogno became critically ill.
“The health officials haven’t been yet able to pinpoint ‘patient zero’,” Angelo Borrelli, head of the national Civil Protection agency, told reporters in Rome.
Mr Borrelli indicated the strategy is to concentrate on closures and other restrictions to try to stem the spread in the country, which already had taken measures early on in the global virus alarm that included banning direct flights from China, Hong Kong, Taiwan and Macau.