The Democrat demanded outgoing President Donald Trump "step up" and repudiate the violence.

Mr Trump, who had urged the demonstrators to march on the Capitol, later called on them to "go home".

A joint session of Congress confirming Mr Biden's election win has resumed after it was suspended by the mayhem.

US Vice-President Mike Pence started the session on Wednesday evening, saying it had been a "dark day in the history of the United States Capitol".

Earlier, the protesters fought their way past police into the complex, shouting and waving Trump and US flags as they roamed the halls, demanding the results of the presidential election be overturned.

In stunning scenes beamed around the world, the invasion sent members of Congress scrambling for cover under their seats as a gunshot rang out and tear gas was fired in the Capitol Rotunda.

A female civilian who was shot inside the US Capitol during the chaos has died, a spokesperson for the Metropolitan Police Department said.

Federal law enforcement official said two suspected explosive devices were found and were both rendered safe by the FBI and Capitol Hill police.

The rampage came as two Democrats won Senate seats in elections in Georgia, which shifted the balance of Congress to their party's effective political control, aiding the passage of Mr Biden's agenda after he is inaugurated on 20 January.

The Democrat, who defeated the Republican president in November's White House election, said the protesters' activity "borders on sedition".

Speaking from Wilmington, Delaware, he also said democracy was "under unprecedented assault".

"I call on President Trump to go on national television now to fulfil his oath and defend the Constitution and demand an end to this siege," he said.

"To storm the Capitol, to smash windows, to occupy offices on the floor of the United States Senate, rummaging through desks, on the House of Representatives, threatening the safety of duly elected officials.

"It's not protest; it's insurrection."

Mr Trump responded in a recorded video on Twitter, repeating his unproven claims of election fraud.

"I know your pain. I know you're hurt," said the president.

"We had an election that was stolen from us. It was a landslide election, and everyone knows it, especially the other side.

"But you have to go home now. We have to have peace."

For the first time, Twitter froze Mr Trump's account. The social media company announced it would keep the account locked for 12 hours, demanding he delete three tweets that it said could stoke violence and threatening "permanent suspension". Facebook and Instagram followed suit.

Earlier on Wednesday, Mr Trump addressed a "Save America Rally" outside the White House.

He had urged supporters to head to the Capitol and said: "Our country has had enough and we will not take it anymore."

Little is known about the woman who died, but disturbing footage from the scene shows her slumped on the ground with blood on her face. Police have not yet released her identity and it is unclear who fired the shot.

Local US media have named her as a San Diego-area US Air Force veteran and Trump supporter, Ashli Babbitt.

The protesters surged up the Capitol steps at around 14:15 local time (19:15 GMT), shoving past barricades and officers in riot gear to penetrate the building.

The mob - some of whom wore body armour - used chemical irritants to attack police, according to Washington Metropolitan Police Chief Robert Contee.

They marched through the building shouting "Where are they?" and chanting "We want Trump".

One climbed on to the Senate dais and shouted: "Trump won that election." Another protester was photographed sitting in House of Representatives Speaker Nancy Pelosi's office with a foot on the table.

Police with guns drawn piled furniture against the doors of the House chamber to keep protesters out.

Members of Congress - including Vice-President-elect Kamala Harris, who is a senator - were told to evacuate the building.

Several thousand National Guard troops, FBI agents and US Secret Service were deployed to help overwhelmed Capitol police.

After an occupation lasting several hours, the sergeant-at-arms - the executive office of the Senate - announced that the building had been secured by law enforcement.

But there was little sign the protesters were heeding Mr Trump's call to go home, despite a citywide curfew declared by the city mayor from 18:00 to 06:00 (23:00 to 11:00 GMT).

Meanwhile, a suspicious device was found outside the Republican National Committee's headquarters, near the Capitol complex, and it was detonated by a bomb squad.

Washington DC Metropolitan Police said five guns were seized. At least 13 people were arrested.

There were also protests at state legislatures in Kansas, Georgia, Utah and on the other side of the country in Oregon and the north-western state of Washington.

By Laura Trevelyan, BBC News, Washington

On the steps of Capitol Hill, hundreds of loyal Trump supporters were packed closely together, as nearby armed police officers kept a watchful eye.

The mood was tense and defiant.

"We're not [expletive] Antifa!" one man screamed at the police, referring to the loose coalition of "anti-fascist" activists that oppose Mr Trump.

Trump loyalists near him waved placards that said "show us the ballots".

"All we want is for the Capitol police to stand down, and surrender the building to us," said one man to news cameras, as he was filmed by other Trump supporters.

The conviction was that the election was stolen from President Trump, and the lawmakers inside the building should do their duty and somehow award the election to him.

Never mind that election officials have certified the results and the courts have thrown out Trump campaign lawsuits alleging fraud because there's no evidence.

A joint session of Congress was being held to certify Mr Biden's election victory on 3 November.

The proceedings are usually brief and ceremonial but Republican lawmakers have been objecting to some results.

For days Mr Trump had also been piling pressure on Mr Pence, who was presiding over the session, to block certification of the result. "Do it Mike, this is a time for extreme courage!" the president tweeted on Wednesday.

But in a letter to Congress, Mr Pence said that he had no "unilateral authority to decide which electoral votes should be counted".

Senate Republican leader Mitch McConnell also definitively broke with Mr Trump in an emotional speech from the chamber floor.

The Kentucky senator warned: "If this election were overturned by mere allegations from the losing side, our democracy would enter a death spiral."

After the session resumed, Senate Democratic leader Chuck Schumer blasted the mob at the Capitol as "rioters and insurrectionists, goons and thugs, domestic terrorists".

He said the president "bears a great deal of the blame".

Senator Kelly Loeffler, who lost her bid for election in Georgia's vote on Tuesday, said she could no longer in good conscience vote against certification as she had originally planned, citing the "abhorrent" invasion of the Capitol.

Political figures across the world expressed shock. UK Prime Minister Boris Johnson condemned "disgraceful scenes" and called for a "peaceful and orderly transfer of power".

German Foreign Minister Heiko Maas said: "Trump and his supporters must accept the decision of American voters at last and stop trampling on democracy."

And former US President George W Bush said in a statement: "It is a sickening & heartbreaking sight. This is how election results are disputed in a banana republic - not our democratic republic."

Former President Barack Obama said in a statement that history would rightly remember the assault on the Capitol as "as a moment of great dishonour and shame for our nation".