The 26-year-old, who has been linked with a move to Real Madrid, fractured his right ankle while on international duty with Belgium on Sunday.
Chelsea said the operation on Monday was "successful", but that Hazard will not return to training for approximately three months.
The new Premier League season is due to begin on 12 August.
Hazard will also miss pre-season games against Arsenal, Bayern Munich and Inter Milan in Singapore from 22-29 July, and the Community Shield against Arsenal on 6 August.
Hazard was instrumental as Chelsea won the Premier League last season, scoring 16 goals in 36 games.
Chelsea forward Eden Hazard will miss the start of the Premier League season after having surgery on a broken ankle.