Everton went ahead in the ninth minute when Seamus Coleman burst into the area and finished with a powerful strike.
Saints substitute Gaston Ramirez equalised in the 71st minute, beating Toffees goalkeeper Joel Robles, who it appeared should have done better in his attempt to save the effort, with a shot from distance.
The three points mean Roberto Martinez's side are - for a few hours at least - up a place to fourth in the table, while Southampton remain ninth.
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Butterflies and nerves of steel were behind the images shared and seen by millions across the globe.
The Duke and Duchess of Sussex thanked everyone who took part in the celebrations on Saturday, Kensington Palace said.