Belgium striker Lukaku headed home Arouna Kone's 22nd-minute cross and swept in from Barkley's astute pass on the stroke of half-time before the England midfielder neatly curled in a third goal six minutes from time.
Victory, Everton's first at St Mary's since 2002, provisionally elevated Roberto Martinez's team to second place in the fledgling standings, two points below Manchester United, who won 1-0 at Aston Villa on Friday.
An impressive seventh last season in manager Ronald Koeman's first campaign, Southampton are without a win after two matches, having drawn 2-2 at Newcastle United in their opening fixture.
West Ham United will look to build on last weekend's surprising 2-0 win at Arsenal when they host Leicester City later on Saturday, while Swansea City, who drew 2-2 at champions Chelsea, entertain Newcastle.
Manchester City host Chelsea in an early clash of title contenders on Sunday, when the visitors will hope to move on from the furore sparked by manager Jose Mourinho's demotion of medics Eva Carneiro and Jon Fearn. – AFP
Original source: Lukaku stars as Everton beat Saints