The Lotus driver, whose wife Marion Jolles met 27-year-old Grosjean as a F1 reporter for French television TF1, confirmed the happy news this week.
"Our son Sacha was born on the 29th of July," he said on Twitter, referring to the very day after he finished sixth in the Hungarian grand prix.
After a wave of up and down form in 2013, Grosjean also qualified a season-best third in Hungary, while many applauded his feisty racing on Sunday.
Asked to name a highlight of 2013, he said: "Well, I would like to say the pass on Felipe (Massa) in Budapest, which at the time I thought may have been the best of my career so far, but of course with the penalty I suppose it's not the same.
"After our performance in the last few races, I honestly believe my first win is now just around the corner," Grosjean added.
Indeed, while earlier it was believed Grosjean's erratic form might cost him his race seat, boss Eric Boullier thinks Lotus has "a future contender for titles" on its hands now.
"It gives me great pleasure to see him learn from those (bad) experiences and to really start delivering the kind of results we've always known he is capable of," he said after Hungary.
"He knows that if he wants to be winning races and fighting for championships then he must deliver this kind of quality drive at every grand prix," Boullier added.
As for his wife and his newly-born son, Frenchman Grosjean told his 165,000 Twitter followers that "All the family is doing very well".