Sir Tom's coffin was carried to Bedford Crematorium by soldiers from the Yorkshire Regiment, where a private service was held on Saturday afternoon with his two daughters, Hannah Ingram-Moore and Lucy Teixeira, his four grandchildren and sons-in-law.
His granddaughter read a poem she helped to write saying he found “happiness in everything”.
The veteran who captured the hearts of the nation raised more than £32 million for the NHS.
He died at Bedford Hospital on February 2 after testing positive for Covid-19.
A World War Two-era C-47 Dakota, part of the Battle of Britain Memorial Flight which operates from RAF Coningsby in Lincolnshire, performed a flypast over the funeral service.
A firing party of 14 fired three rounds in unison, while a bugler sounded The Last Post at the end of the private service.
Sir Tom asked that his epitaph read: "I told you I was old", in reference to comedian Spike Milligan's famous epitaph: "I told you I was ill".
Once Covid-19 restrictions permit, the family will inter Sir Tom's ashes in Yorkshire, with his parents and grandparents in the Moore family plot.
The celebrant conducting the funeral of Captain Sir Tom Moore said: "It's quite incredible to think that 163 countries donated to (Sir Tom's) fundraiser - that's almost the whole world.
"As wonderful as we think our NHS is, people from other countries really aren't going to be interested in our health, so it seems obvious to me that they were really investing in Captain Tom and the values he stood for.
"He was a proud British veteran and a gentleman, he lived in a multi-generational environment, not only would that have kept him young, but also symbolises the importance of family to him.
"What sacrifices did he and his peers make in defence of our freedom, a man with a strong moral compass, a strong work ethic, a sense of pride and an indomitable spirit.
"He serves as an inspiration to us all to never give up and always stay strong knowing tomorrow will be a better day."