Messi confirmed the arrival of Ciro on Instagram, publishing a photograph of the newborn holding his father's hand and revealing that everything had gone well. Mom and he are very well.
Messi had pulled out of Barcelona's squad for Saturday's La Liga match at Malaga to be present for the birth.
The player's wife, Antonella, is believed to have delivered their third son on Saturday.
Ciro is a younger brother to Messi's sons Mateo and Thiago, about whom Roccuzzo regularly posts on social media.
Messi was replaced by Yerry Mina for the game at La Rosaleda, with Barcelona citing "personal reasons" for his absence.
Valverde's side have opened up an eight-point lead at the La Liga summit and remain well on course to return the Spanish top-flight crown to Camp Nou.