The signal-caller was in tears as he left the field on a cart after doing the damage when he was tackled by Giants defensive back Logan Ryan with just over six minutes remaining in the third quarter.
Dallas on Monday confirmed Prescott's operation had gone to plan.
A statement from the team said: "The Cowboys announced Monday morning that Dak Prescott's ankle surgery was successfully completed Sunday night, and he is expected to be released from the hospital [on] Monday.
"Prescott suffered a right ankle compound fracture and dislocation in the third quarter of the Cowboys' win over the Giants.
"The injury was splinted with a sterile dressing, and he was transported to a local hospital for surgery to wash out the wound and fix the fracture."
Cowboys owner and general manager Jerry Jones said of Prescott on Sunday: "He is an inspiration to everyone he touches. He has all of our love and support.
"And we have no doubt that he will return to the position of leadership and purpose that he brings to our team."
Prescott, who was replaced by Andy Dalton in the win over the Giants, started the 2020 season on a one-year exclusive franchise tender after he and the Cowboys were unable to strike a deal on a long-term contract.
The 27-year-old will earn approximately $31.4million for the season before becoming an unrestricted free agent, unless the Cowboys opt to again place the franchise tag on him – a move that would count over $37m towards the 2021 salary cap.