Mr Brown was speaking as a host of senior Labour figures past and present joined friends and 60 family members for the non-religious service at London's Golders Green Crematorium.
Mr Foot died at his Hampstead home on March 3.
About 160 gathered in the austere red-brick West Chapel to hear Mr Brown declare "he was one of the most elegant of orators, one of the greatest parliamentarians ever - so many talents".
The Prime Minister paid tribute to "a life lived in the service of the greatest of progressive causes" and added: "We mourn today for one of our own."
Mr Brown said the depth of loss at Mr Foot's passing went far beyond the chapel and "throughout our country and beyond". He went on: "He graced the country and for those of us privileged to know him, he graced all of our lives."
Mr Brown recalled taking his son John to see Mr Foot, who promised him "a little present".
It turned out to be a first edition of Jonathan Swift's Gulliver's Travels.
The premier closed his contribution by quoting a passage from Bunyan's Pilgrim's Progress: "I do not repent me of what trouble I have been at, but arrive where I am, my marks and scars I carry with me".
Neil Kinnock, who succeeded Foot as Labour leader in 1983, hailed him as "brilliant and brave" in thought, word and deed.
Mr Foot's coffin was carried into the black-and-white marble-floored chapel draped in red roses and the scarf of his beloved Plymouth Argyle to the strains of Verdi's Va Pensiero, the Chorus of the Hebrew Slaves.
Great-nephew John Foot - son of the late investigative journalist Paul Foot - introduced the service.
Squeezed into pews beside the family were Labour's Deputy Leader Harriet Harman, Chief Whip Nick Brown, Climate Change Secretary Ed Miliband, former Prime Minister's wife Cherie Blair, Wales Secretary Peter Hain, former Downing Street communications director Alastair Campbell, Foreign Office minister Baroness Kinnock and former union leaders Lord (Bill) Morris and Rodney Bickerstaffe.
Ex-ministers Frank Dobson, Mark Fisher and Michael Meacher were joined by veteran backbencher Dennis Skinner and arts broadcaster and Labour peer Lord (Melvyn) Bragg.
John Foot told the gathering: "Michael had a huge effect on all our lives and we will miss him terribly."
He paid special tribute to his great-uncle's carer Jenny Stringer - who had promised the former Labour leader's late wife Jill Craigie 10 years ago that she would look after him.
Following Mr Brown's tribute, the ex-editor of Tribune magazine Mark Seddon said of Mr Foot's passing: "It has deeply moved so many people up and down the land."
And he stressed the late Mr Foot would have been urging Labour on to victory at this year's general election, saying: "He had real hope for what we can be again."
Mr Foot had been associated with Tribune since its birth and remained honorary editor-in-chief until his death.
He was also a founding member of the Campaign for Nuclear Disarmament, whose former head Bruce Kent told the gathering: "Michael was no pacifist. He was deeply committed nevertheless to world peace and to the elimination of all nuclear weapons."
He said Michael Foot had considered them illegal, barbaric and counter-productive.
Mr Foot's period as Labour leader from 1980-83 saw him widely ridiculed for his perceived shabby dress and chaotic hairstyle.
Mr Kent said: "Despite the abuse and contempt heaped on him by the right-wing press, I have never heard Michael indulge in personal bitterness or criticism."
He added: "If you want to see his monument, look around you at all those whose hopes were kept alive not only because of CND but because of his principled stand."
Michael Foot's step-daughter Julie Hamilton praised him as "more than just a step-father, he was a wise and wonderful grandfather to my four children and also my friend".
Mr Foot first entered Parliament as an MP for Plymouth in 1945 before switching to become MP for Ebbw Vale - later renamed Blaenau Gwent.
Close friend and former vice-chairman of Plymouth Argyle Peter Jones told how for his 90th birthday the ex-Labour leader had officially been registered with the Football League as a player for the club, with the squad number 90.
His official pen-portrait described Mr Foot as "a vigorous left-winger who brings plenty of experience".
Mr Jones continued: "Plymouth gave Michael his roots and his passion." He recalled the city's role in bringing Foot and his late wife together, as she was producing a film about its regeneration after it was devastated in the Second World War.
Another great-nephew, Tom Foot, declared: "It is a terrible thing waking up in a world without Michael Foot."
The last of the seven speakers, Lord Kinnock, said: "In thought and in word and in deed, Michael Foot was brave and brilliant."
Lord Kinnock continued: "Michael loved the House of Commons not as a place of trappings and tradition, but as the supreme instrument for bringing the powerful to account and for gaining redress for the powerless."
He then launched into an impersonation of Mr Foot hurling some of his famous jibes at Conservative adversaries Norman Tebbit and Michael Heseltine and SDP founder David Owen.
He also disclosed that Mr Foot's love of dogs "produced a rapport with the Queen".
Lord Kinnock said it had been Mr Foot's "most self-sacrificing act" to agree to become leader of the Labour Party aged 67 at the urging of friends.
He went on: "I was not among those who appealed to him to stand. I thought even if he won he would be submitting himself to endless torture."
Mr Foot was indeed subjected to "agonies", he said.
"Even Michael's gigantic capacity for forgiveness was stretched by all of that," said Lord Kinnock.
But he added Mr Foot's "raw courage saved the Labour Party from terminal division".
And he continued: "Michael's religion was humanity, his country was the world. He is dead now but the memory of him endures to illuminate all our lives. His great soul goes striding on."
Lord Kinnock then led the gathering in an emotional rendition - joined by the Prime Minister - of The Red Flag, sung by at least one veteran socialist with a raised clenched fist.
Mr Foot's body was committed to the music of Rossini's La Gazza Ladra. Those inside the chapel then filed out beneath its lilac-tinted rose window to mingle in the last of the early spring sun with around 100 mourners who had gathered outside to watch the service on TV monitors.
Mr Brown, accompanied by his wife Sarah, chatted to mourners before heading off back to Westminster.
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