Matt Hancock sought to reassure the public following panic-buying in some areas, with supermarkets seeing their shelves cleared of essentials such as toilet roll and paracetamol.
It comes as an older patient, reported to be a woman in her 70s, became the first person in the UK to die after being diagnosed with coronavirus.
Speaking on BBC's Question Time, Mr Hancock said: "The Government has supplies of the key things that are needed, and, within the food supply, we are absolutely confident that there won't be a problem there.
"And, crucially, we are working to makes sure that if people are self isolating, they will be able to get the food and supplies that they need."
He said there was "absolutely no need" for individual people "to go round buying more than they need."
He added: "The very, very strong advice from the scientists, from the medics, is that people should not go buying more than they need."
Mr Hancock also told anyone self-isolating with coronavirus to stay away from their own family members as much as possible, and to wipe down shared surfaces such as in bathrooms.
He said: "People should try to self-isolate from their families, not only go home, try not to go out shopping, definitely don't use public transport, but within your own home you should also try to self-isolate."
He said that, as the father of three children, he understood that "can be difficult and some people have caring responsibilities", but people should try to keep to themselves as much as possible.
Meanwhile, 142 people from the UK - including 121 passengers and 21 crew - are on the Grand Princess cruise ship currently quarantined off the coast of California.
Princess Cruises said it was following advice from the US Centres for Disease Control and Prevention (CDC) and samples were collected on Thursday from 45 people deemed to be at risk.
It said in a statement: "Following guidelines received from the CDC, all guests have been asked to stay in their staterooms while test results are pending.
"Guests are receiving meal deliveries in their staterooms by room service, and additional television and movie options have been added to in-room programming."
The cruise ship made a round trip from San Francisco to Mexico last month, after which a cluster of coronavirus cases were identified.
After the ship docked in San Francisco on February 21, thousands of passengers boarded and thousands disembarked. The ship then sailed to Hawaii.
Overall, there are 3,533 people currently on board the Grand Princess, including 2,422 guests and 1,111 crew, of 54 nationalities.
On Thursday evening, the Royal Berkshire Hospital in Reading said the older patient who died had been "in and out of hospital" for other reasons but was admitted on Wednesday evening and tested positive.
It came during a day of several key developments, with the number of UK cases of Covid-19 surging to 116 - more than double the total 48 hours earlier - and Downing Street warning it was "highly likely" the virus would spread "in a significant way".
Health chiefs also said people diagnosed with coronavirus who show only "very minimal" symptoms should self-isolate at home rather than in hospital, while new advice was issued to travellers returning to the UK from anywhere in Italy that they should self-isolate if they develop symptoms.
Following the news of the UK death, Prime Minister Boris Johnson offered his sympathy to the family of the patient, while England's Chief Medical Officer, Professor Chris Whitty, said he was "very sorry" to report the news.
He added: "We believe they contracted the virus in the UK and contact tracing is already under way."
Last week, a British tourist who had been on board the Diamond Princess cruise ship, which was quarantined in Japan, became the first UK citizen to die from the virus.
At a press conference on Thursday evening, Prof Whitty revealed that 18 people have so far recovered from coronavirus in the UK and 45 are being treated at home.
"We have moved to a situation where people have very minimal symptoms and we think they are clinically safe and they are able to self-isolate. We think it is actually safer for them, as well as more pleasant, if they can self-isolate in their own homes," he said.
"Anybody who needs hospitalisation will be hospitalised."
He had earlier warned that critical care beds in the NHS could come under intense pressure during a coronavirus epidemic, and told MPs the UK had mainly moved to the delay stage of tackling the virus.
This could include measures such as school closures, encouraging greater home working, and reducing the number of large-scale gatherings.
But the PM stressed it was important not to "fire your shots too early" in escalating measures to tackle the illness.
In terms of national prevention measures, Mr Hancock said Prof Whitty told him that stopping flights into the UK would only delay the arrival of the disease "by a matter of days".
He added that halting flights to the UK would also make it "much harder" to get medicines into the country because they are produced abroad.
Prof Whitty has said half of all coronavirus cases in the UK are most likely to occur in just a three-week period, with 95% of them over a nine-week period.