The home side flew out of the traps quickly and took the lead in the seventh minute when Nemanja Matic flicked on a corner and Costa lashed home at the far post for his seventh goal of the season.
Antonio Conte’s side doubled their lead just past the half-hour mark when Pedro played Hazard through and the Belgian rounded Kaspar Schmeichel and slotted home.
The Foxes were poor throughout and offered little in attack, with their best chance of scoring coming through David Luiz when the Chelsea man prodded the ball against his own post.
The game was up for the visitors 10 minutes from time when substitute Nathaniel Chalobah produced a sublime flick to set up Moses, who fired in his second goal of the season and in the process condemned Claudio Ranieri’s side to their fourth league defeat of the season, one more than they lost in their 2015-16 title-winning campaign.
Southampton have announced plans for a two-week tour of China as part of their pre-season preparations for the 2018-19 season.