Baby Eric, who is named after Simon's late father, was born in New York's Lennox Hill Hospital, where stars including Beyoncé have previously welcomed their children.
"Very happy to say Eric was born at 17.45pm. Healthy and handsome," tweeted the X Factor supremo. "Lauren in great shape. Thanks for all of your kind messages. Named after my dad."
In the run up to the birth, American socialite Lauren had been spending a lot of quality time with her older son Adam, who is seven. His father is her ex-husband Andrew Silverman.
The 37-year-old also previously spent New Year's Eve enjoying a 'babymoon' with her millionaire beau on Slipstream, the yacht he rents every year. The couple were seen strolling together on a beach in Barbados, hand in hand, looking the picture of contentment.
At one point during their break the mogul showed how much he was looking forward to fatherhood when he stopped to enthuse over a toddler. Simon even asked the baby's mother whether he could hold her child.
Despite his newfound happiness, the 54-year-old has no plans to marry the mother of his child for the present time as he exclusively revealed to HELLO! in October.
Lauren finalised the details of her divorce on 4 December after ten years of marriage.
The news of her pregnancy with Simon came out in July.
Initially the pop executive refused to comment saying: "I really have to be sensitive here because there are a lot of people's feelings involved."
Later he opened up saying it was "surreal" to see the scan and admitting he felt "paternal". One fatherly task that Simon does plan to delegate is dealing with dirty nappies. "I'm not doing that, 100 per cent absolutely not going near that," he said.
Sleepless nights are unlikely to be a problem either. Warned by comedian Harry Hill that babies tend to leave you exhausted, he joked: "Can't I just get a nanny?"