"Beau Biden was an original. He was a good man. A man of character. A man who loved deeply and was loved in return," the president said.
Beau Biden died at the end of last month from brain cancer aged 46.
He was seen as a rising star of US politics but suffered from health problems in recent years.
Beau Biden had intended to run for Delaware state governor in 2016.
About 1,000 people - including Bill and Hillary Clinton and numerous other top politicians - attended the funeral at a Roman Catholic church in Wilmington, the largest city in the state of Delaware.
Vice-President Joe Biden led a procession into the church with his family at the beginning of the service.
Mourners heard President Obama describe Mr Biden as a public servant who learned through early tragedy what mattered most and as a result decided upon living "a life of meaning" that would inspire those around him.
"He was a scion of an incredible family," the president said, "who brushed away the possibility of privilege for the harder, better reward of earning his own way."
The president described Beau Biden as a son, a father, a soldier and a politician who refused to take short cuts in his determination to serve his country and others.
He said that a "cruel twist of fate" killed Beau Biden's mother and infant sister in a car crash four decades ago and left Beau - three years old at the time - and his younger brother Hunter in hospital.
The accident happened shortly after Mr Biden won election to the Senate in 1972
Coldplay singer Chris Martin was a soloist at the service - he volunteered to perform after hearing that Beau Biden liked the band.
Mr Biden was diagnosed with brain cancer in August 2013 and underwent treatment that was initially successful. However, the cancer recurred earlier this year.
Barack Obama has delivered an emotional eulogy at the funeral of former Delaware Attorney-General Beau Biden, the son of US Vice-President Joe Biden.