Sir David was stabbed multiple times at his constituency surgery in Leigh-on-Sea in Essex on Friday.
The Metropolitan Police said there was a potential link to Islamist extremism.
A 25-year-old British man was arrested at the scene on suspicion of murder, and police said they were not seeking anyone else over the death.
As part of the investigation, officers are carrying out searches at two addresses in the London area, the Met said.
The force believes the man acted alone but inquiries into the circumstances of the incident are continuing.
The man is in custody in Essex, police added.
Government sources have told the BBC he is a British national who, from initial inquiries, appears to be of Somali heritage.
Sir David, 69, who represented Southend West, was holding a constituency surgery – where voters can meet their MP and discuss concerns – at Belfairs Methodist Church on Friday when he was attacked at 12:05 BST.
Essex Police Chief Constable BJ Harrington said Sir David was “simply dispensing his duties when his life was horrifically cut short”.
Counter-terrorism officers are working with Essex Police and the Eastern Region Specialist Operations Unit.
“The fatal stabbing in Leigh-on-Sea has tonight been declared as a terrorist incident, with the investigation being led by Counter Terrorism Policing,” the Met said.
“The early investigation has revealed a potential motivation linked to Islamist extremism.”
Officers are appealing for anyone with any information or with footage from CCTV, dash cams, or video doorbell, to contact them.
Home Secretary Priti Patel has asked police forces to immediately review security arrangements for MPs, saying the killing represented “a senseless attack on democracy itself”.
She added that “questions are rightly being asked about the safety of our country’s elected representatives”.
Sir David had been an MP since 1983 and was married with five children. He is the second serving MP to be killed in the past five years, following the murder of Labour MP Jo Cox in 2016.
Prime Minister Boris Johnson described him as “one of the kindest, nicest, most gentle people in politics”.
Security concerns routine in 21st Century politics
We expect the MPs we elect to see us in person, not to hide behind Parliament’s ornate gates and wood-paneled walls.
That demand is met gladly by the vast majority of MPs.
But, increasingly, the job has been accompanied by abuse, intimidation – and risk for MPs and their staff.
One member of the cabinet told me today: “Everyone has had a threat… everyone has had frightening moments.”
Dealing with harassment, coping with security concerns and reporting those concerns to the police, is sadly routine in politics in the 21st Century.
It is inevitable in the coming days that there will be calls for a kinder atmosphere at Westminster, and cooler heads in real life, and online.
It is not, however, inevitable that anything at all will change.
House of Commons Speaker Sir Lindsay Hoyle told BBC Two’s Newsnight police were contacting all MPs to check on their security and reassure them.
He went ahead with his own constituency surgery on Friday evening, saying it was essential MPs retained their relationship with their constituents.
“We have got to make sure that democracy survives this,” he said.
But Conservative MP Tobias Elwood – who came to the aid of a stabbed police officer during a terror attack in Westminster in 2017 – told the BBC he would recommend MPs temporarily stop having face-to-face meetings with constituents.
“You can move to Zoom… you can actually achieve an awful lot over the telephone,” he said on Radio 4’s World Tonight.
And Kim Leadbeater, the sister of Mrs. Cox and MP for Batley and Spen, said her partner had asked her to stand down from her role following Sir David’s death.
Who was Sir David Amess?
A Conservative backbencher for nearly 40 years, Sir David entered Parliament in 1983 as the MP for Basildon.
He held the seat in 1992 but switched to nearby Southend West at the 1997 election.
Raised as a Roman Catholic, he was known politically as a social conservative and as a prominent campaigner against abortion and on animal welfare issues.
Tributes have been paid to Sir David from across politics and within his local community.
Prime Minister Boris Johnson said he had an “outstanding record of passing laws to help the most vulnerable”, adding “we’ve lost today a fine public servant and a much-loved friend and colleague”.
Father Jeff Woolnough, the parish priest at nearby St Peter’s Catholic Church, led a mass on Friday evening in memory of Sir David, who he called “Mr. Southend”.
He described him as a “great, great guy” and said faith communities had “lost their greatest supporter”.
Southend councilor John Lamb said Sir David was “a very good, hard-working constituency MP who worked for everyone”.
And Labour leader Sir Keir Starmer said it was a “dark and shocking day”, adding that the country had “been here before” with the death of Jo Cox.