Sir Tom's coffin, draped in a union flag, was carried to the crematorium on Saturday by soldiers from the Yorkshire Regiment while a Second World War-era C-47 Dakota performed a flypast.
Sir Tom's 100th birthday celebrations previous year included a Spitfire flypast and he was photographed punching the air as it went past.
A version of the song Smile, recorded by singer Michael Buble, was to be played at the funeral.
Defence Secretary Ben Wallace said: "In national emergencies ordinary people do extraordinary things and inspire us all to pull together to overcome adversity".
"Few will have heard of Sir Tom before this crisis but his contribution and example now lives on in us all".
In a statement, his daughters said Captain Sir Tom had spoken "openly about his death and his funeral" over the past year and had "wondered out loud if perhaps the interest in him over the last 12 months would mean we would need to have more Victoria Sponge cakes available for the extra guests".
Sir Tom, who raised more than £32 million for the NHS with a sponsored walk of his garden in the first lockdown, served with the Duke of Wellington's Regiment during the Second World War.
The regiment later merged with two others from Yorkshire, becoming the Yorkshire Regiment, and Sir Tom was made an honorary colonel last August.
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Soldiers performed ceremonial duties at the service for the 100-year-old, whose charity walk inspired the nation and raised nearly £33 million ($60m).
This was followed by a firing party of 14 each firing three rounds in unison before a small service got under way.
The service, at Bedford Crematorium, was attended by eight members of Sir Tom's immediate family - his two daughters Hannah Ingram-Moore and Lucy Teixeira, four grandchildren and his sons-in-law - all wearing masks.
The family set up an online book of condolences that gathered thousands of messages. The soldiers then marched out and left his close family for the service, to the sound of Moore singing "You'll Never Walk Alone" in a charity single he recorded with Michael Ball.
His family has asked the public not to attend his funeral and "continue to support the NHS by staying at home".
Captain Sir Tom Moore's daughter, Lucy, said her father would have been "chuckling" after plans were unveiled.
He had asked for his epitaph to be "I told you I was old, " inspired by comedian Spike Milligan whose gravestone says "I told you I was ill".
His ashes were later to be interred in a family plot in his native Yorkshire in northern England, with his parents and grandparents.
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