The child was born at a London hospital, the spokesperson said, adding that mother and baby are doing very well.
“The PM and Ms Symonds would like to thank the fantastic NHS maternity team,” the spokesperson said.
Figures from across the British political spectrum congratulated the couple after the news emerged.
“Great to hear Downing Street is getting a new resident,” Chancellor Rishi Sunak said on Twitter.
Lindsay Hoyle, Speaker of the House of Commons, said: “On behalf of everyone in the House of Commons, may I say congratulations to the Prime Minister and Carrie Symonds on the birth of their son. Such happy news amid so much uncertainty — 2020 is certainly a year they will never forget.”
Johnson returned to work on Monday, after becoming the first world leader to fall ill with coronavirus in March.
The UK leader spent three nights in intensive care in early April, before his condition eventually improved.
Symonds previously said on Twitter that she had also experienced coronavirus symptoms.
Symonds, 32, has had a prominent career working in politics and was part of the campaign team that helped to get Johnson re-elected as mayor of London in 2012.
The British Prime Minister has four children from his second marriage.
The last time a child was born to a sitting prime minister was in 2010, when David Cameron and his wife, Samantha Cameron, welcomed their daughter Florence.
This is a breaking story, more to follow…