The man whose voice was known to millions around the world for his commentary on the Edinburgh Tattoo and more than 300 state occasions died in St Columba's Hospice on 18 April, aged 82.
The Rev Dr Fiona Douglas, who led the service at Warriston crematorium, said that for many people Mr Fleming's "voice of fire and velvet" had been the voice of the nation.
"Tom was a public figure, yet he was deeply private. He was enormously humble in a way only the great can be humble."
She said he had prepared the funeral service himself and left clear instructions – "no sad and mournful gathering".
And the Rev Ron Ferguson called Mr Fleming "a commanding actor with an amazing stage presence", a perfectionist, a consummate professional and national institution.