"Zaniolo underwent reconstruction of the anterior cruciate ligament in his right knee, along with repairs to the external meniscus," Roma said in a statement.
It added "the surgery... was successful and Zaniolo will begin his recovery on Tuesday".
The 20-year-old midfielder was stretchered off the pitch at the Stadio Olimpico in tears during Sunday's 2-1 defeat to Serie A champions Juventus.
Recovery for such an injury takes between four and six months, and Zaniolo could miss Euro 2020.
"Thank you all for the support and affection you have shown me during the past few hours," he wrote on Instagram beside a photo of him in his hospital bed.
"I can't wait to get back out there. Forza Roma!"
Italy play the opening match of Euro 2020 against Turkey at the Stadio Olimpico on June 12 before facing Switzerland and Wales in the group stages.
Zaniolo has scored twice for the Azzurri in his five appearances and had been expected to be a key player in Roberto Mancini's side.
He has also scored six goals in 23 competitive games this season for Roma.
"Come on Nicolo! I'm waiting for you on the pitch more determined than ever," Italy coach Mancini wrote on Twitter on Monday.