The former England rugby star revealed the news in his podcast, The Good, The Bad and The Rugby.
A spokeswoman for the couple said: “Zara and Mike Tindall are happy to announce the birth of their third child, Lucas Philip Tindall.”
The baby, who is a new great-grandson to the Queen, was born on Sunday 21 March, weighing 8lbs 4oz.
The Tindalls, who live on the royal Gatcombe estate in Gloucestershire, already have two children – Mia, seven, and Lena, two – and have been married since 2011.
The baby was born at home, Mike Tindall revealed on his podcast.
After talking about Saturday’s matches he said: “Sunday got even better because a little baby boy arrived at my house.”
Tindall said his son was born on the bathroom floor after the couple were unable to make it to hospital.
“Arrived very quickly. Didn’t make it to hospital. On the bathroom floor,” he said.
Tindall described how he rushed to get a mat for his wife.
“So yeah, it was running to the gym, get a mat, get into the bathroom, get the mat on the floor, towels down, brace, brace, brace.”
The baby was born at 6pm on Sunday, he said.
Mia and Lena were elsewhere for the day because Zara had been having contractions through the night.
Lucas is the Queen and the Duke of Edinburgh’s 10th great-grandchild, but is only 22nd in line to the throne.
Mike Tindall had joked about wanting a boy this time around.
Buckingham Palace said the Queen and the Duke of Edinburgh are “delighted” at the news and “look forward to meeting their 10th great-grandchild when circumstances allow”.
The birth comes just a few weeks after Zara’s cousin Princess Eugenie had her son August on 9 February.