Kareena, daughter of veteran actors Randhir Kapoor and Babita, delivered the baby on Tuesday morning at the Breach Candy Hospital here.
A statement released on behalf of the actors read: "We are very pleased to share with you all the wonderful news about the birth of our son: Taimur Ali Khan Pataudi, on December 20, 2016.
"We would like to thank the media for the understanding and support they have given us over the last nine months, and of course especially our fans and well wishers for their continued affection. Merry Christmas and a happy new year to you all...With love, Saif and Kareena."
Kareena and Saif, who have worked together in films like "Tashan", "Kurbaan", "Agent Vinod", LOC Kargil" and "Omkara", got married in October 2012.
Saif, the son of veteran actress Sharmila Tagore and the late Mansoor Ali Khan Pataudi, was earlier married to actress Amrita Singh and shares two children with her - daughter Sara and son Ibrahim.
He is doing "extremely well" after becoming a father again, says his sister Soha Ali Khan.
"Delighted to welcome Taimur Ali Khan Pataudi into the world. Both mother and son are healthy and doing well. Father is doing extremely well," Soha tweeted.
Kareena's father Randhir told IANS: "The entire family is overwhelmed and is welcoming the new member in our family.... Thanks for all the blessings."
Her best friend Amrita Arora, who has been part of Kareena's closest circle throughout her pregnancy, posted on Instagram: "My darling, welcome to the best club everrrrr... 'Mommy club'. Love you."
Taimur -- which in Persian language means iron -- has already been dubbed as "mini Nawab" by film fraternity members.
Filmmaker Karan Johar tweeted: "My Bebo had a baby boy! I am so so happy! Taimur Ali Khan."
Actress Sonam Kapoor posted: "Congratulations my darling Bebo and Saif! Taimur Ali Khan Pataudi is going to be the apple of everyone's eye... Mini Nawab."
Filmmaker Satish Kaushik said: "Congrats to my favourite heroine and person Kareena Kapoor Khan and Saif Ali Khan for the arrival of baby boy Taimur Ali Khan."
Actor Tusshar Kapoor commented: "Even more glorious life with God's blessing in hand! Cheers to Saif and her both and peace and everlasting joy to the little prince."