Lukaku, who hit 10 goals in the Premier League last season, scored halfway through the first half with a superb header from six yards, before finishing a sweeping move with Barkley on the stroke of half time for 2-0.
Barkley’s fine finish rounded off the win with six minutes remaining, giving Everton their first three points at St Mary’s since 2002, and leaving Ronald Koeman’s Southampton without a victory so far this season.
Everton fans made their feelings known to owner Bill Kenwright before the match after a banner was seen flying over St Mary’s reading: “Kenwright and Co, time to go”, but the occasion belonged to the Toffees.
Saints striker Graziano Pelle went close with a header six yards out from Dusan Tadic’s cross, before Saido Mane forced a fine save out of Tim Howard from a tight angle on the left.
But just 43 seconds passed from the Everton goalkeepers’ stop and the visitors taking the lead in the 22nd minute, Lukaku heading home from Arouna Kone’s pin-point right-wing cross following a speedy break.
Barkley spurned a fine opportunity to double Everton’s lead, hitting the ball straight at Maarten Stekelenburg after being found in acres of space central to goal, before Tom Cleverley fluffed the rebound wide.
But Barkley had a big hand to play in Everton’s second, collecting Lukaku’s pass on the halfway line and slipping it back to the Belgian 16 yards out, Lukaku duly applying the finish low into the bottom left corner.
Southampton came out fighting in the second half, but Pelle missed two decent opportunities to halve the deficit. First, his fierce first-time volley fell straight into the hands of Howard, before he miscued a free header from a corner with 20 minutes remaining.
But Everton rounded off a fine counter-attacking performance late on, Barkley collecting a Seamus Coleman pass on the edge of the area and curling past Stekelenburg into the far right corner.