Ayoze Perez and Jamie Vardy both hit hat-tricks as Brendan Rodgers' side moved second with a huge win that continued their superb start to the season.
The rout just equalled Manchester United's 9-0 demolition of Ipswich at Old Trafford in March 1995.
Leicester are just five points behind league leaders Liverpool, who host Tottenham Hotspur today, after a historic display that sank the Saints, who drop into the relegation zone on goal difference, at a soaking wet St Mary's Stadium.
"We said at half time, keep your speed in the game and we have to punish them," said Rodgers.