Charlie Austin's goal 41 seconds in was the winner, making for an unhappy return for Ronald Koeman to St. Mary's Stadium.
Man United and West Ham finished in a 1-1 stalemate that saw Jose Mourinho sent off for a second time this season.
Alexis Sanchez scored a brace as Arsenal beat Bournemouth 3-1 at the Emirates Stadium.
Eddie Howe says despite gifting Arsenal their first goal, Bournemouth looked to be in control the majority of the game.
Arsene Wenger says Arsenal came out in the second half much calmer and were able to defeat a tricky Bournemouth side.
Ronald Koeman complimented his side's defence despite losing 1-0 to Southampton and said Everton had good chances to score.