Speaking ahead of a private funeral service, the Prime Minister said the MP's colleagues were "absolutely devastated" by the news.
Sir David, a father-of-five, was stabbed to death while holding a constituency surgery at Belfairs Methodist Church in Leigh-on-Sea, Essex, on October 15.
Mourners paid their respects to the Southend West Tory MP during a private ecumenical service at St Mary's Church in Prittlewell.
The devout Catholic's friend and colleague Mark Francois delivered a eulogy, praising Sir David's service to his constituents and his sense of humour.
The Rayleigh and Wickford MP told those gathered: "Our electors employ us to represent them in a contract renewable every few years.
"We work for them and not the other way around, and no-one was ever more conscious of that than David Amess.”
Former Conservative MP Ann Widdecombe, a friend of Sir David's, read a statement on behalf of the Amess family, asking people to "set aside hatred" and urging tolerance.
Sir David's coffin, draped in a union flag, was carried by pallbearers from Southend Fire Service.
After the church service, they carried the coffin to a horse-drawn hearse for a procession around Southend.
Hundreds of people gathered outside Southend's Civic Centre to pay their respects as the hearse, led by four black horses, paused in front of it.
A requiem mass will be held at Westminster Cathedral in London on Tuesday, where a message from the Pope will be shared.
Earlier, Mr Johnson said he and his Cabinet became aware of Sir David's death while taking part in a technology away day in Bristol.
He told the BBC: "Everybody was absolutely devastated.
"We were all sitting around. I then got called out and got given the news.
"I had to go back in to tell Cabinet colleagues, many of whom had known David for decades, and I'm afraid several colleagues broke down in tears because it was just an appalling piece of news.
"I think we were also very shaken by the implications of what had happened, and the fact his life had been tragically ended in the way that it was.
"I've spoken to members of his family and they've had a very tough time, as you can imagine."
He called Sir David's legacy "multifarious", adding: "He campaigned on endometriosis, on fuel poverty, on the registration of driving instructors, on children with learning disabilities - he has a tremendous, natural sympathy for people who needed help. He showed what a backbencher can do.
"He has a big legacy of things he actually achieved as a backbencher.
"He was a valiant campaigner for a long time for leaving the European Union, and he was ultimately successful in that, and he has the permanent memorial of having transformed Southend into a city.
"His legacy really is that he will be a kind of exemplar of what a constituency MP can do."
Mr Johnson insisted the last thing Sir David would want now "would be for that vital free exchange between MPs and constituents to be interrupted or closed down as a result of the appalling circumstances of his [death]".