His body was carried from the Casa Rosada, the government palace in the historic centre of Buenos Aires, to the Jardin de Bella Vista cemetery, where only relatives and close friends attended the ceremony.
Numerous fans lined up along the route of the funeral procession to pay their last respects to the football great.
Thousands of people had earlier come to the government palace to bid farewell to Maradona, who died at the age of 60 on Wednesday.
According to TV broadcasters TyC and ESPN, Maradona's body arrived at the Casa Rosada around 5 am (0800 GMT). During the night a private ceremony was held for family and friends.
Members of the public were able to attend from 6 am to 4 pm. But many of them arrived at Plaza de Mayo, in front of the Casa Rosada, hours before to sing loudly in Maradona's honour.
Around 30 minutes before the doors were due to open, a huge crowd was already lined up and a strong police presence was on hand.
In small groups of 20 people were allowed to walk into the government headquarters to say their last goodbye. Visibly affected, many threw flowers, jerseys and other small gifts over Maradona's coffin.
Some queued for hours for the chance to briefly pass by the coffin, which was adorned with football shirts and a national flag.
Many crossed themselves and there were shouts of "Thank you, Diego" or "I love you, Diego."
Later in the day, some fans forced entry into the Casa Rosada, according to television reports.
They climbed over the fence at the entrance, apparently because they feared they would not be let in to see his coffin and pay their last respects.
Police used tear gas to disperse the crowd, while inside the palace, Maradona's coffin was moved to a safer place in another room.
Clashes meanwhile broke out near the presidential palace. Angry fans hurled stones and bottles at police, who responded with rubber bullets and water cannons.
Media reports said Maradona suffered a heart attack on Wednesday and could not be revived.
Maradona's lawyer Matias Morla said on Twitter he will ask for an investigation into the death. He alleged that medical help was on hand at Maradona's home but there were huge delays, including after an ambulance was called.
"It's inexplicable that for 12 hours my friend didn't get attention from health personnel. The ambulance took more than 30 minutes to arrive," he wrote.
"This fact cannot be overlooked and I'll ask for an investigation. As Diego used to tell me: you're my soldier, act with no mercy.
"To describe Diego in this moment of deep sadness: he was a good son, the best player in history and an honest person."
Maradona, who led Argentina to World Cup glory in 1986, underwent surgery for a bleed on the brain earlier this month.
The forward is widely regarded by football fans as one of the greatest players to ever play the game.