Both mother and baby are doing very well, their spokesman said.
The couple announced at the end of February that Ms Symonds was pregnant and they had got engaged.
“The PM and Ms Symonds would like to thank the fantastic NHS maternity team,” their spokeswoman said.
Sky News understands that the PM will take paternity leave later in the year because he wants to focus on dealing with the COVID-19 crisis.
Foreign Secretary Dominic Raab will stand in for Mr Johnson at Prime Minister’s Questions later.
The PM’s father Stanley Johnson said he was “absolutely delighted” by the birth of his grandson, while messages of support from across politics poured in following the news.
Health Secretary Matt Hancock posted on Twitter: “So thrilled for Boris and Carrie. Wonderful to have a moment of unalloyed joy!”
Chancellor Rishi Sunak tweeted: “Great to hear Downing Street is getting a new resident. Congratulations to @BorisJohnson and @carriesymonds on the birth of their son.”
Labour leader Sir Keir Starmer said: “Wonderful news. Many congratulations to Boris Johnson and Carrie Symonds.”
And Scottish First Minister Nicola Sturgeon wrote: “Some good news – sending congratulations to Carrie and the PM. And wishing health and happiness to the wee one.”
Ms Symonds announced she was pregnant in February in a post on her Instagram account.
“I wouldn’t normally post this kind of thing on here, but I wanted my friends to find out from me,” she wrote.
“Many of you already know, but for my friends that still don’t, we got engaged at the end of last year, and we’ve got a baby hatching early summer. Feel incredibly blessed.”
Earlier this month Ms Symonds told how life got “very dark” at times while her husband-to-be spent a week in St Thomas’ Hospital with coronavirus, including three nights in intensive care.
Mr Johnson, 55, and 32-year-old Ms Symonds made history as the first unmarried couple to officially live together in Downing Street when they moved in last July.
The baby is Ms Symonds’ first child, while Mr Johnson already has four grown-up children with his second wife Marina Wheeler; Lara, Milo, Cassie and Theodore.
The PM also has a daughter with art consultant Helen Macintyre.
There has been speculation that Mr Johnson has fathered another extra-marital child.
Asked how many children he had during the general election in November, he told LBC: “I love my children very much but they are not standing at this election. I’m therefore not going to comment on them.”
The new arrival is the third baby born to a serving prime minister in recent history.
Tony Blair’s wife Cherie gave birth to son Leo in May 2000, three years after her husband’s first election victory, and David Cameron and wife Samantha welcomed daughter Florence in 2010.
The last babies born to prime ministers before Leo and Florence arrived more than 150 years ago.