"The surgical procedure undergone today... was perfectly successful," Roma said in a statement, adding that the forward "will begin rehabilitation in the coming days."
The 2022 World Cup winner with Argentina picked up the injury during Roma's 1-0 league win against Cagliari earlier this month.
Roma did not say for how long Dybala would be sidelined, but with only nine matches left in the current Serie A campaign, the 31-year-old is almost certain not to play again until next season.
Dybala's absence will be a big blow to Roma, who, since Claudio Ranieri's return as coach in November, have gone from being relegation candidates to the form team in Italy's top flight.
He has scored eight times and set up four more in all competitions, with Roma fighting for Champions League football as they sit four points behind fourth-placed Bologna.