Riyad Mahrez opened the scoring after just 11 minutes when he picked up the ball deep in the Southampton half, charged forward and fired a ferocious effort from distance into the bottom right-hand corner.
The Foxes then doubled their lead six minutes later, Christian Fuchs's effort from the outside the area bouncing off Steven Davis and falling for Okazaki, who duly buried the ball beyond Fraser Forster from close range.
With seven minutes left in the first half, Leicester made it three as Harry Maguire controlled Mahrez's free kick and played the ball across the face of goal, where Andy King arrived unmarked at the back post to slide the ball home.
Any hopes that the Saints had of an unlikely comeback were snuffed out on 69 minutes when Cedric Soares gifted possession to Mahrez, who combined with Jamie Vardy to work the ball across the face of goal, where Okazaki was on hand to rifle home.
The result sees Mauricio Pellegrino's 11th-placed side slip to four points behind 10th-placed Everton, while Leicester's fourth league win in a row keeps them eighth in the table.
Southampton (4-2-3-1): Forster; Soares, van Dijk, Yoshida, Bertrand; Romeu (Gabbiadini 45'), Lemina; Tadic (Ward-Prowse 83'), Davis, Boufal (Redmond 62'); Austin
Leicester City (4-2-3-1): Schmeichel; Simpson, Maguire, Morgan, Fuchs; Ndidi, King; Mahrez (Albrighton 72'), Okazaki, Chilwell (Ulloa 83'); Vardy (Gray 72')