As they rampaged through a busy night-life hub around London Bridge on Saturday night, the trio wore what looked like suicide vests which turned out to be "hoaxes", said Mark Rowley, head of counter-terrorism policing.
"We believe that six people have died in addition to the three attackers shot dead by police," Rowley said, adding it was being treated as a "terrorist incident."
Following reports a fourth attacker was on the loose, Rowley said: "Our current belief is that there were three attackers but this is still early on."
More than 30 casualties were rushed to hospitals in the area, according to the London Ambulance Service.
The injured included a British Transport Police officer who was one of the first responders on the scene and was stabbed in the face and leg.
The attack came just five days ahead of a general election and just minutes after the end of the Champions League final between Real Madrid and Juventus in an area teeming with bars where many fans were watching the football on television.
"They were stabbing everyone. They were running up and going 'This is for Allah'," a man called Gerard told the BBC, adding that he had seen the assailants stabbing a girl and had tried to confront them.
Another witness called Eric told the BBC he had seen three men come out of the white van after hitting pedestrians and thought they were going to help.
But instead they "started kicking them, punching them and took out knives. It was a rampage really," he said, adding that he also heard a shout of: "This is for Allah".
Britain is on high alert only 12 days after a suicide bomber killed 22 people at a concert in Manchester, northwest England and ahead of Thursday's general election, in which security is a major theme.
It is the latest in a string of attacks to hit Europe, including in Paris, Berlin and Saint Petersburg.
Echoes of Westminster attack
Witnesses described the van speeding into several pedestrians on London Bridge and then one or more knife-wielding men sprinting towards bars packed with revellers enjoying a Saturday night out.
One witness described seeing as many as five assailants and several people said they were ordered to stay inside pubs and restaurants by police and eventually had to come out with their hands on their heads.
Italian photographer and documentary maker Gabriele Sciotto, who was watching the football at the Wheatsheaf pub in Borough Market, said he saw three men shot just outside the pub.
In his picture, a man wearing combat trousers, with a shaved head and what looked like a belt with canisters attached to it could be seen on the ground with two more bodies behind him.
"In two or five seconds, they shot all the three men down," Sciotto told the BBC.
Police said the three men were shot by a police armed response team within eight minutes of receiving the first call at 2108 GMT.
Dozens of emergency vehicles could be seen and a wide area around London Bridge was cordoned off as two helicopters hovered overhead.
The attack had harrowing echoes of the one on London's Westminster Bridge when 52-year-old British Muslim convert Khalid Masood rammed his car into pedestrians before crashing into the barriers surrounding parliament and then stabbing a police officer to death.
Masood was shot dead by a ministerial bodyguard.
'We are with you'
Prime Minister Theresa May, who is due to hold an emergency cabinet meeting on Sunday, confirmed the "terrible incident in London" was being treated as "a potential act of terrorism."
"Our thoughts are with those who caught up in these dreadful events," she said in a statement.
Facebook activated its safety check function for people in London to let their loved ones know they are safe.
US President Donald Trump offered his help, tweeting "WE ARE WITH YOU. GOD BLESS!"
Trump also seized the occasion to plug his thwarted ban on travellers from six mainly Muslim countries.
French President Emmanuel Macron meanwhile said France was "more than ever at Britain's side."
US pop star Ariana Grande, whose concert in Manchester was the scene of last week's fatal terror attack, tweeted: "Praying for London".
'Her throat had been cut'
Witnesses on London Bridge reported seeing a van mounting the pavement and hitting pedestrians.
jpegMpeg4-1280x720"There was a van that crashed into the fences on London Bridge. And then there was a man with a knife, he was running. He came down the stairs and went to the bar," Dee, 26, who was visibly in shock and declined to give her last name, told AFP.
Alex Shellum at the Mudlark pub near the scene of the attack said a woman had come into the bar "bleeding heavily from the neck".
"It appeared that her throat had been cut," he told the BBC.
Husband and wife Ben and Natalie told BBC radio they were outside Borough Market when they witnessed a stabbing.
Ben said: "We saw people running away and then I saw a man in red with a large blade, at a guess 10 inches (25 centimetres) long, stabbing a man, about three times.
"It looked like the man had been trying to intervene, but there wasn't much he could do. He was being stabbed quite coldly and he slumped to the ground."
Ben said the man then walked towards another pub. He said they saw a metal chair being thrown towards the man.
"Then we heard three gunshots, definitely gunshots, and we ran."
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