Kaunda, who was also a hero of the struggle against white rule in southern Africa, died last month at the age of 97.
Some of his relatives had wanted him buried at his farm next to his late wife Betty, with whom he had 10 children.
But the burial proceeded without incident at Embassy Park, a special cemetery dedicated for the country’s leaders.
His children paid glowing tribute to Kaunda, portraying him as a fighter and teacher whose attributes should be emulated.
“Go well, you fought the good fight of faith, you have finished the race. We will miss you so much,” said one of his daughters, Musata Banda.
“You wanted three more years to get to a hundred, but you will celebrate your 100th birthday in heaven.”
President Edgar Lungu declared Kaunda’s birthday, April 28, as a national holiday in honour of the first president.
Zambia had already announced three weeks of mouring after his death on on June 17 at the Maina Soko military hospital in Lusaka.
Hundreds of mourners attended a church service before the burial, including Mozambique’s ex-president Joachim Chissano, Zambia’s former president Rupiah Banda and Hakainde Hichilema, the head of the main opposition United Party for National Development (UPND).
Leader of Kaunda’s United National Independence Party (UNIP), Trevor Mwamba, urged Zambia to honour Kaunda by holding peaceful elections next month.
The country goes to presidential and legislative election on August 12.
Kaunda’s first-born son, Panji, called for peaceful polls.
“We shall move together as ‘one Zambia, one nation’,” he said, reiterating his father’s motto.
“We have elections, let us preach peace. Stop the violence that has been witnessed.”
Outside the cathedral, Lusaka resident Alex Mwale, 19, also seized the moment to appeal for peaceful voting.
“Kaunda as our father stood for peace. This is what our politicians should learn from this man. He was in the same class with great men like Nelson Mandela,” said Mwale.