Stopping in the hospital's chapel, the pope was given a basket containing handwritten messages from the children.
"Thank you for your dreams and prayers that you have put in this basket," the pope said, according to a Vatican statement. "Let's entrust them together to the Lord, who knows them more than anyone."
Heading into the various wards during the visit of nearly three hours, the 77-year-old pontiff waved away photographers, saying: "I'm here for the patients."
The Vatican-owned hospital, Bambino Gesu (Baby Jesus), is Europe's largest paediatric research centre, with a staff of 2,600 serving some 27,000 patients each year.
Pope John XXIII began the tradition of papal visits to Bambino Gesu at Christmastime in 1958.