Matt Hancock said it was “very important” for the staff, patients and society that they get the jab.
Meanwhile, a hospital trust has revealed that only 36.8% of its black staff have had the vaccine.
Everyone in the top four groups has had a jab offer and 15 million people have had a first dose, the government says.
The rollout in England has officially been expanded to those aged 65 and over and younger people in at-risk groups.
In a bid to encourage more social care staff to have the jab, Mr Hancock told BBC Breakfast: “It is the right thing to do. It is very important that you come forward and take up this vaccine. It’s important for you, it’s important for your patients and of course it’s important for the whole of society.”
Analysis from the University Hospitals of Leicester NHS Trust has revealed lower proportions of black and Asian staff being vaccinated.
It found that 70.9% of white staff had come forward, compared with 58.5% of South Asian staff and 36.8% of black staff.
Prime Minister Boris Johnson urged those eligible to receive the vaccine to come forward.
“These jabs, these vaccines are safe, they are efficacious, they will help protect you against disease and against death and they’re a wonderful thing to have – they help protect you, your family, your neighbours,” Mr Johnson said.
Prof Stephen Powis, national medical director for NHS England, said he would be talking to clinical leaders from various ethnic groups “to work out together how we can encourage everybody to take up the vaccine”.
The uptake among care workers isn’t surprising – they tend to come forward in lower numbers for the flu jab.
It illustrates the need to continue reaching out to those who have not taken up the offer.
The fear is some of the most vulnerable and deprived communities have the lowest levels of uptake, which would widen inequalities in the future.
Today, though, marks an important milestone. But that is all it is.
Most Covid deaths have come in the over-70s, but just over half of hospitalisations have.
It means there are a significant number at risk in the 50 to 70 age group as well as younger adults with health conditions. They have been promised a jab by the end of April.
Achieving that will not be easy. From March people will start needing second doses, which means the rate at which first doses are given is expected to tail off.
As always, everything depends on supply. Ministers are confident in the supply of the Pfizer and AstraZeneca vaccines and by Easter the deliveries of a third jab made by Moderna should arrive. There can be no let up.
Meanwhile, Mr Hancock said that there was still “some way to go” before easing lockdown.
He said: “At the latest count there are still over 23,000 people in hospital with Covid – that’s more than in the April peak.”
It is 69 days since the UK began its mass vaccination programme – and six weeks since the prime minister set out the pledge to offer a first dose to the four priority groups by 15 February.
That meant vaccinating all elderly care home residents and their carers; everyone over 70; all frontline health and social care workers; and everyone with a condition that makes them extremely vulnerable to the virus – a total of 14.6 million people.
Mr Johnson has hailed the vaccine rollout as an “extraordinary feat” and said the country can go into the next stage “with great confidence”.
But while the UK’s vaccination rate is third in the world, behind only Israel and the United Arab Emirates, there have been reports that some people in priority groups – including those who are housebound – are still waiting for their first dose.
And a small number of the 15 million people vaccinated will have been outside these initial priority groups as some NHS regions had already begun giving jabs to younger people.
As well as the over-65s, the next group to be offered the vaccine is the estimated 7.3 million people aged 16 to 65 with a range of health conditions that put them at risk from the virus.
The rollout will then be extended to the remaining over-50 age groups, with the aim of offering a first dose to another 17.2 million people before the end of April.
Meanwhile, the government has confirmed that millions of people whose asthma is under control will not be prioritised for the vaccine.
Mr Hancock said that 90% of the over-70s had accepted the vaccine and one in four adults were now starting to receive protection against Covid-19.
The Department of Health said 88% of the UK’s coronavirus deaths occurred in the priority groups who have now been offered the vaccine.
With the expectation that increasing numbers of people will be protected from serious illness, ministers will this week finalise plans on how England’s coronavirus measures can be eased.
The government has said it will take “cautious” steps, rejecting calls from Conservative MPs to commit to abolishing all Covid-19 restrictions by the end of April.