Video report by ITV News UK Editor Paul Brand
The prime minister said he takes "full responsibility" for the huge death toll, adding: "We truly did everything that we could, and continue to do everything that we can, to minimise loss of life and to minimise suffering."
The Covid-19 death toll surpassed the grim milestone on Tuesday, with the government confirming another 1,631 fatalities.
It brings the UK's total coronavirus death toll to 100,162 - one of the highest in the world.
Mr Johnson said when the pandemic is over the government will "make sure we learn the lessons and reflect and prepare" - possibly a nod to those calling for an inquiry.
ITV News Political Editor Robert Peston analyses how the UK has reached 100,000 deaths
At the start of the pandemic Chief Scientific Adviser Sir Patrick Vallance said keeping coronavirus deaths below 20,000 nationally would be a "good outcome".
On Tuesday afternoon Chief Medical Officer Chris Whitty said that "unfortunately" there will be "additional deaths to add to that very sad total" of 100,000 deaths.
Offering some encouragement, the prime minister said Covid-19 vaccines will eventually free the UK from the pandemic and "put us on a path to recovery".
The prime minister, speaking at a Downing Street press conference offered his "deepest condolences" to those who have lost loved ones, as he urged people to stick with the lockdown restrictions.
"It's hard to compute the sorrow contained in that grim statistic: The years of life lost, the family gatherings not attended and for so many relatives the missed chance to even say goodbye," he said.
He added: "And I'm deeply sorry for every life lost. What I can say is that the government will continue to do everything we can to minimise life lost as we go forward."
On lockdown restrictions in place around the UK, Professor Whitty warned: "I think we need to be careful we do not relax too early".
He acknowledged that Covid-19 levels are dropping marginally, but the number of people in hospital with the virus is still "incredibly high".
ITV News Health Editor Emily Morgan says deaths will continue to rise significantly, as we are only just beginning to hear of deaths of people who contracted Covid-19 at end of December
Prof Whitty warned that "the number of people dying per day will come down relatively slowly over the next two weeks".
Health Secretary Matt Hancock said his thoughts are with "each and every person who has lost a loved one - behind these heart-breaking figures are friends, families and neighbours".
He said Covid-19 vaccines offer a "way out" of the pandemic, "but we cannot let up now".
"We sadly still face a tough period ahead. The virus is still spreading and we’re seeing over 3,500 people per day being admitted into hospital."
Labour leader Sir Keir Starmer said the figure represents a "national tragedy" and is a "terrible reminder of all that we have lost as a country".
He added: "To all those that are mourning, we must promise to learn the lessons of what went wrong and build a more resilient country. That day will come and we will get there together.
“But for now we must remember those that we have lost and be vigilant in the national effort to stay at home, protect our NHS and vaccinate Britain.”
Liberal Democrats leader Ed Davey called for an independent inquiry into how the UK reached such a high death toll, saying there must be justice for bereaved families.
Reaching 100,000 deaths comes just over one year since health authorities in China reported that a 61-year-old man in Wuhan had died from a “new type of coronavirus”.
Just months after, in March 2020, the UK saw its first death of a person who tested positive for the virus - a woman aged in her 70s who had underlying health conditions.
Little did we know back then that the death toll in the United Kingdom would reach 50,000 eight months later in November.
And with Covid-19 evolving and new strains forming in the UK's south-east as well as South Africa, the death toll doubled in less than half the time to reach 100,000 in January 2021.
When did most deaths occur?
January 2021 has overtaken April 2020 to become the UK's bleakest month during this pandemic.
On January 20, 1,820 Covid deaths were reported within 28 days of a positive coronavirus test - the peak number of deaths in a single day.
How does the UK compare to the rest of the world?
The UK has the world's worst daily death rate for coronavirus related fatalities, according to research by Oxford University.
Statistics revealed the UK to have the worst seven-day average of new daily Covid-19 deaths per million people - a rate of 16.54 per 1,000,000.
Britain knocked the Czech Republic - which had the highest daily death rate since January 11 - off the top spot when it published the death figures on January 17.
Why are deaths reported as ‘died after testing positive for Covid-19’?
When a person dies with the virus in their body, it must be reported they had the infection - it could be the major cause, a contributory factor or they may die of something else unrelated.
Covid-19 deaths are reported weekly by the Office for National Statistics (ONS) and daily by the government (these are the figures reported by ITV News every day) incorporating the UK as a whole, and by all health bodies of the the four nations individually.
The ONS numbers are higher because they include deaths where Covid-19 was recorded as a cause of death on the death certificate, whether or not there was a lab-confirmed test, but where the doctor believes Covid-19 to have caused or contributed to the death based on the deceased’s symptoms. There is also no time frame on when the test was taken to the time of death.
Watch the press conference in full: