The UK is the first country in the world to approve a Covid vaccine, with the regulator stressing "the most rigorous scientific assessment" was carried out before doing so.
So far the UK has ordered 40 million doses of the Pfizer/BioNTech jab - enough to vaccinate 20 million people.
An initial order of 800,000 vaccines will be in the UK from next week.
Health Secretary Matt Hancock said it is a "day to remember, frankly, in a year to forget".
"We can see the way out of this but we're not there yet, so let's keep our resolve, keep doing our bit to keep people safe until science can make us free."
Details of who will be first to get vaccinated - and how - have been outlined by the country's leading vaccination experts.
Residents in care homes and their carers are first on the list with those aged over 80 next in line as well as frontline health and social care workers.
The MHRA also gave details of how the vaccine was approved, and what measures were taken to ensure it was safe.
Dr June Raine, of the MHRA medicines regulator, told the briefing the Pfizer vaccine met "rigorous high standards" and stressed "the public’s safety" had always been priority.
Health secretary Hancock, updating MPs on the vaccine development, acknowledged the Pfizer vaccine is "particularly challenging to deploy" because it must be stored at -70 degrees.
Asked how many people could be vaccinated by January, Mr Hancock said: "While today brings more certainty, it doesn't end all uncertainties - while we have 800,000 (doses) that have now passed the batch testing, the total number that will be manufactured over this timeframe is not yet known because it's all dependent on a manufacturing process which is complicated itself."
But difficulties around production and supply did not dampen the health secretary's spirits.
"It's no longer if there's going to be a vaccine, it's when, he said.
"In our battle against the virus, help is on its way. Today is a triumph for all those who believe in science, a triumph for ingenuity, a triumph for humanity."
Special storage containers will be used to ship the vaccine, with GPS trackers to remotely monitor its temperature. While Pfizer has designed a suitcase-sized container that will keep the doses at ultra-low temperatures for up to 10 days using dry ice.
'It will be ready to roll out from next week' Matt Hancock tells ITV News
Mr Hancock said there would be "three modes of delivery" - hospitals, vaccination centres, and community rollout (including GPs and pharmacists).
Prof Wei Shen Lim, who sits on the Joint Committee on Vaccination and Immunisation, admitted there may be some "operational constraints" in delivery of the vaccines to care homes.
But added that he hoped that in the first phase of the vaccine programme 99% of the UK's most clinically vulnerable would be vaccinated.
"While we know that this vaccine protects you from getting ill with Covid – we don’t yet know how much it stops you transmitting Covid until we roll it out broadly," he told Sky News.
"The plan is to get this rolled out, according to the clinical prioritisation that the advisers will set out."
Mr Hancock said he had spoken to his "counterparts in the devolved nations" on Wednesday morning "to make sure that we are all ready to roll out this vaccine … from early next week."
The health secretary also suggested Brexit was part of the reason the UK has become become the worldwide to have a clinically authorised vaccine.
Matt Hancock tells ITV's Good Morning Britain the vaccine would not have been approved if it was not clinically safe
He told Times Radio: "The MRHA has done a great job of working with the company to look at that data as it’s come through and do things in parallel, rather than one after the other as they normally would, that’s the first reason.
"The second reason is because, while until earlier this year we were in the European Medicines Agency (EMA), because of Brexit we’ve been able to make a decision to do this based on the UK regulator, a world-class regulator, and not go at the pace of the Europeans, who are moving a little bit more slowly."
But June Raine from the medicines regulator MHRA told a Downing Street briefing it was because of "provisions under European law which exist until January 1" that the UK had been able to "authorise the supply" of the Pfizer vaccine.
"Our speed or our progress has been totally dependent on the availability of data in our rolling review and the independent advice we have received," she said.
What you need to know about each vaccine:
The University of Oxford and AstraZenica vaccine is up to 90% effective, can be stored at fridge temperature and is the cheapest, costing just £4 a dose. The UK government has ordered 100 million doses, with four million ready to go.
The Pfizer/BioNTech vaccine has up to 95% efficacy when it comes to immunisation, requires to be stored at -70C degrees and will cost £15 a dose. The UK has ordered 40 million doses, 10 million of which will be available by the end of the the year.
The Moderna vaccine is up to 94.5% effective, requires to be kept at -4C and will cost between £24 and £28 per dose. The UK has five million doses on order.
All three vaccines require two separate inoculations to be effective.
A Department of Health and Social Care (DHSC) spokesman said the approval "follows months of rigorous clinical trials and a thorough analysis of the data by experts at the MHRA".
It said the regulatory group had concluded the Pzifer/BioNTech "met its strict standards of safety, quality and effectiveness."
The DHSC spokesperson urged the public to continued to "abide by the necessary restrictions in their area".
Watch the vaccine expert's Pfizer briefing in full
Sir Simon Stevens, chief executive of the NHS in England, said the vaccination programme would be the “largest-scale vaccination campaign in our country’s history”.
Reacting to the announcement, Scotland's First Minister Nicola Sturgeon said it the “best news in a long time”.
England’s chief medical officer, Professor Chris Whitty, said that, despite the positive vaccine news, “we can’t lower our guard yet”.
He tweeted: “The independent regulator authorised the first vaccine for use against Covid-19.
“This is excellent news and a step towards normality. It will take until spring until the vulnerable population who wish to are fully vaccinated. We can’t lower our guard yet.”
On ITV's Good Morning Britain, Health Secretary Matt Hancock says he'd happily take a Covid-19 vaccine 'because you've got to show leadership in these things'
She tweeted: “@scotgov ready to start vaccinations as soon as supplies arrive.”
Wales' First Minister Mark Drakeford tweeted: “Thank you to everyone who has worked so hard to make this a reality. Our vaccine programme is ready to go, but the impact won’t be seen nationally for some months."
Labour leader Sir Keir Starmer tweeted: “Thank you to all those involved in this wonderful news – from the brilliant scientists to the trial volunteers.
"We must now ensure vaccines are rolled out safely but swiftly across the country.”
“The government needs to put in place a plan to roll out the vaccine rapidly. They failed on PPE, failed on testing and failed on track and trace. We cannot afford another failure.”
Nadim Zahawi, the newly-appointed minister responsible for overseeing the vaccination rollout, tweeted: “Major step forward in the fight against Covid 19 today.”