The van struck people after mounting the pavement, outside Muslim Welfare House near Finsbury Park Mosque on Seven Sisters Road in the city, in the fourth terrorist attack in four months in the UK.
"It is a reminder that terrorism, extremism and hatred take many forms; and our determination to tackle them must be the same whoever is responsible," May said in a statement at Downing Street after chairing a meeting of the government's emergency COBRA committee.
May later paid a visit to Finsbury Park mosque, close to the scene of the attack, where she reiterated that "there has been far too much tolerance of extremism over many years".
A 47-year-old man, the driver of the van, was detained by members of public at the scene and then taken to hospital and later arrested on suspicion of the commission, preparation or instigation of terrorism including murder and attempted murder.
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The driver drove the van into worshippers close to Muslim Welfare House in Finsbury Park as they were gathered to help an elderly man who had collapsed. The elderly man later died, but it is not clear if this was a result of the attack. Nine other people were taken to hospital.
"This is being treated as a terrorist attack and the Counter Terrorism Command is investigating. At this early stage of this investigation, no other suspects at the scene have been identified or reported to police, however the investigation continues," Metropolitan Police deputy assistant commissioner Neil Basu had said earlier.
Met Police Commissioner Cressida Dick said the incident in north London was "quite clearly an attack on Muslims", and the community would now see more police, including armed officers, in the area, "particularly around religious establishments".
"There is now an ongoing investigation by our Counter Terrorism Command to establish why this attack was carried out. London is a city of many faiths and many nationalities. An attack on one community is an attack on all of us. Terrorists will not succeed in their attempts to divide us and make us live in fear," she said.
"It is home to a multitude of communities that together make London one of the greatest cities on earth. Diverse, welcoming, vibrant, compassionate, confident and determined never to give in to hate," she said in her statement.
"This was an attack that once again targeted the ordinary and the innocent going about their daily lives - this time British Muslims as they left a Mosque having broken their fast and prayed together at this sacred time of year," she added.
May called for security at mosques and other places of worship in the lead up to Eid Al-Fitr to reassure the community.
"This was an attack on Muslims near their place of worship. And like all terrorism, in whatever form, it shares the same fundamental goal. It seeks to drive us apart; and to break the precious bonds of solidarity and citizenship that we share in this country. We will not let this happen," she said.
London mayor Sadiq Khan said more officers were being deployed at places of worship to reassure the community.
"Terrorism is terrorism... While this appears to be an attack on a particular community, like the terrible attacks in Manchester, Westminster and London Bridge it is also an assault on all our shared values of tolerance, freedom and respect," he said, adding that the city was grieving and still recovering from the Grenfell Tower fire tragedy in west London.
Many of the victims are believed to have just left evening prayers at the Muslim Welfare House after breaking the Ramadan fast.
Eyewitness Abdul Rahman said the driver said he wanted to "kill all Muslims" and the Muslim Council of Britain (MCB) said it was a "violent manifestation of Islamophobia" and called for extra security around mosques.
"The van was driving towards us to try and basically hit us at speed. When he got arrested, he was taunting, saying, 'I'd do it again, I'd do it again'," another witness said.
"We expect the authorities to increase security outside mosques as a matter of urgency," Khan said.
UK home secretary Amber Rudd described it as an appalling incident and said new funding for security at religious sites had recently been arranged.
"We have a places of worship fund which we announced last summer, which is there to protect places of worship like mosques. We will make sure that we do all we can to reduce these sort of attacks," she said.
Kacimi said when he did so, the driver said to him, "I've done my bit."
Mohammed Mahmoud, the imam of Muslim Welfare House who has been hailed a "hero" told reporters later today that a passing police van was flagged down after the attack.
"We told them the situation - there's a man, he's restrained, he mowed down a group of people with his van and there is a mob attempting to hurt him and if you don't take him then, God forbid, he might be seriously hurt. We pushed people away from him until he was safely taken by police," he said.
Forensics officers are examining a white van, which has the label of Pontyclun Van Hire.
The London Ambulance Service said it was called to the incident at 12.15 am (local time) today morning.
"We sent over 60 medics to the scene, including ambulance crews, advance paramedics and specialist responses teams and an advance trauma team from London's Air Ambulance," said the service's Deputy Director of Operations Kevin Bate.
This marks the fourth terrorist incident in the UK in four months, after attacks in Westminster, Manchester and on London Bridge.
Eight people were killed and 50 injured on June 3 when three Islamist terrorists drove into pedestrians on London Bridge and stabbed people at nearby restaurants and bars in Borough Market.
Two weeks earlier, a suicide bomber killed 22 people at a concert by American pop singer Ariana Grande in Manchester in northern England.
On March 22, a man drove a rented car into pedestrians on Westminster Bridge in London and stabbed a policeman to death before being shot dead. The attack killed five people.