The Duchess of Cambridge was seen drying her eyes after the Chief Rabbi spoke of the build up to the Shoah and the rounding up of Europe's Jews.
He said: "We stood together after we were rounded up, not knowing where we were being sent to, this was our fate we had no choice."
But defiantly the Chief Rabbi told those in attendance they should "never tire in our efforts to stand together to confront antisemitism, racism, hate speech and demonisation."
Mr Johnson said he felt "a deep sense of shame" that antisemitism continues today.
Addressing those gathered at Central Hall, he said: "I feel a deep sense of shame that here in Britain in 2020 we seem to be dealing with a resurgence of the virus of antisemitism and I know that I carry a responsibility as Prime Minister to do everything possible to stamp it out."
He vowed to ensure the Shoah is not forgotten and also spoke of the "sanctity" of the survivors' testimonies.
The letter read out by Prince William described how Princess Alice, the Duke of Edinburgh's mother, had asked to be informed about the fate of the "Cohen" family.
After learning their lives were in danger, she hid Rachel Cohen and her young daughter in her home, despite the danger to her own well being.
The letter said: "It was thanks to the courageous rescue of Princess Alice that the members of the Cohen family were saved.
"The members of the Cohen family left the residence three weeks after liberation, aware that by virtue of the Princess's generosity and bravery had spared them from the Nazis."
The royal couple also lit candles in memory of those killed, which were used to share lights until 75 candles were burning, to mark the 75 years since Auschwitz was liberated.
Mala Tribich, who survived Ravensbruck and Bergen-Belsen, told the ceremony: "I feel like it is my duty to speak for all those whose voices were silenced and tell of those dark days in Europe.
"By speaking out it is my greatest hope that something positive will be handed down to future generations.”
She appeared on stage to recall how she was taken to Sweden after liberation, before finally being reunited in the UK with her brother Ben – now Sir Ben Helfgott.
Sir Ben was seated next to the royal couple throughout the ceremony.
Labour's Mr Corbyn was seated next to two trustees of HMDT.
Survivor Arek Hersh, who was sent into a work camp in Poland aged 11, told how he was ordered to carry out forced labour at Auschwitz-Birkenau, a death march to Buchenwald and finally Theresienstadt.
Ian Forsyth, a wireless operator with the 15th/19th King’s Royal Hussars in the Royal Armoured Corps, wept as he told of the death of an 11 year-old boy he saw after liberating Bergen-Belsen in April 1945.
“I hope people can realise how far mankind can sink if they are not careful,” he said. “We must learn that freedom is for everyone. We’re not all the same but that should be celebrated. "
In a video, Olivia Marks-Woldman, chief executive of HMDT, said she was pleased the royal couple had been able to attend the ceremony alongside members of the UK's political, civic and faith leadership.
She said: "At a time when we know identity-based hostility is increasing, it is heartening to see so many people stand together - both at the UK ceremony and at more than 10,000 local activities around the country.
"Holocaust Memorial Day is an important opportunity for us all to learn from genocide, for a better future."
The JC was invited into the private room after Monday's event to watch as the Duke and Duchess of Cambridge meet with survivors - including Arek Hersh.
At one stage, the Duke was heard discussing the reaction of German guards to the scenes of brutality in the death camps.
He said: "I am surprised there were not more acts of kindness."
The Duchess spoke with survivors to discuss their efforts today to meet and speak with schoolchildren to tell them of the horrors of the Shoah.
She praised their "interventions" and also heard how the impact of seeing survivors discuss the dark events of the past can help young people in the determination to speak out over injustice in the world today.
Last week, Prince Charles had visited visited Princess Alice's tomb on the Mount of Olives in Jerusalem.
She was honoured as “Righteous Among the Nations” in Israel for sheltering the Cohen family.
Arriving for Monday afternoon's ceremony, Prince William had told Ms Marks-Woldman, Sir Ben and Martyn Atkins, Superintendent of Central Hall, "We were talking this morning about how you carry on this message for future generations. We will do our best."
This year also marks 25 years since the Srebrenica genocide, in which Bosnian Serb troops and paramilitaries led by Ratko Mladic attacked the town, murdering some 8,000 men and boys.
