As Avci was carried away, he was put on a video call with his parents, who showed him his newborn baby.
"I had completely lost all hope. This is a true miracle. They gave me my son back. I saw the wreckage and I thought nobody could be saved alive from there," his father said.
An exhausted Avci was later reunited with his wife Bilge and daughter Almile at a hospital in Mersin.
Aid organisations say the survivors will need help for months to come with so much crucial infrastructure destroyed.
In neighbouring Syria, already shattered by more than a decade of civil war, the bulk of fatalities have been in the northwest, an area controlled by insurgents who are at war with President Bashar al-Assad - a conflict that has complicated efforts to aid people affected by the earthquake.
The sides clashed overnight for the first time since the disaster, with government forces shelling the outskirts of Atareb, a rebel-held town badly hit by the earthquake, the Syrian Observatory for Human Rights reported on Friday.
Thousands of Syrians who had sought refuge in Türkiye from their country's civil war have returned to their homes in the war zone - at least for now.
Neither Türkiye nor Syria have said how many people are still missing following the quake.
For families still waiting to retrieve relatives in Türkiye, there is growing anger over what they see as corrupt building practices and deeply flawed urban development that resulted in thousands of homes and businesses disintegrating.
One such building was the Ronesans Rezidans (Renaissance Residence), which keeled over in Antakya, killing hundreds.
"It was said to be earthquake-safe, but you can see the result," said Hamza Alpaslan, 47, whose brother had lived in the apartment block.
"It's in horrible condition. There is neither cement nor proper iron in it. It's a real hell."
Türkiye has promised to investigate anyone suspected of responsibility for the collapse of buildings and has ordered the detention of more than 100 suspects, including developers.
The United Nations on Thursday appealed for more than $1 billion in funds for the Turkish relief operation, and has launched a $400 million appeal for Syrians.
