An energetic Toffees side took the lead after just 15 minutes when Leighton Baines and Morgan Schneiderlin combined on a free kick to play in Theo Walcott, who finished coolly beyond a stranded Alex McCarthy.
Brazilian forward Richarlison made it 2-0 to the Blues just after the half-hour mark, before Danny Ings notched after the interval as Southampton ran Everton close on what turned out to be a pleasing day for new boss Silva.
Relive all of the action with our minute-by-minute updates below.
EVERTON STARTING XI: Pickford; Coleman, Keane, Holgate, Baines; Gueye, Schneiderlin; Walcott, Sigurdsson, Richarlison; Tosun
SOUTHAMPTON STARTING XI: McCarthy; Soares, Stephens, Hoedt, Bertrand; Ward-Prowse, Romeu, Lemina, Redmond; Austin, Ings