Both sides' Premier League campaigns began with 2-2 draws, although Saints' result at Newcastle brought a far more positive reaction than the Toffees' underwhelming comeback draw at home to promoted Watford.
Some disgruntled fans called for chairman Bill Kenwright and the board to leave Everton in a banner flown over St Mary's, but those frustrations were forgotten for 90 minutes at least as Lukaku's double and a classy Ross Barkley effort secured an impressive 3-0 win.
It was some way to end a six-match winless run on the south coast and dampened the feel-good factor around St Mary's, where late owner Markus Liebherr was remembered five years after his passing.
Everton dashed Saints' hopes of marking the occasion with a win as Lukaku nodded home a superb opener midway through the first half - a goal one home fan may well have applauded given the Belgium international made his way into the stands during the warm-up to check on her after his wayward shot hit her.
The £28million striker continued to boss the hosts' backline and added another as half-time approached
Barkley was the architect with a fine pass and added extra gloss to the scoreline as the match came to a close, curling home a fine third in front of England manager Roy Hodgson.
The game will have made uncomfortable viewing for Southampton manager Ronald Koeman, back on the sidelines after missing the trip to Newcastle following Achilles surgery.
The hosts started well enough, with Graziano Pelle, Sadio Mane and Dusan Tadic having efforts of varying quality before Matt Targett forced Tim Howard into his first save of the match in the 17th minute.
The England Under-21s left-back, deputising for the injured Ryan Bertrand, was already in Michael Oliver's notebook by that point having hacked down Lukaku in full flight.
Barkley's free-kick was deflected behind for a corner and Arouna Kone's shot from the resulting corner was blocked, but Martinez's side managed to break the deadlock in the 22nd minute.
Saints won a corner after Howard smartly denied Mane but Everton broke at speed from it, with Kone played down the right and crossing for Lukaku to head in off the underside of the bar.
It was a sucker punch Saints failed to respond to, with James McCarthy seeing a cross-shot kept out before Tom Cleverley wasted a gilt-edged chance to score after Barkley's initial effort was saved.
Everton were not made to pay for that missed opportunity, though, as Barkley played through Lukaku, who confidently slotted past Maarten Stekelenburg as half-time approached.
Saints were all too easily being carved open and Koeman acted at the break, bringing on defensive midfielder Oriol Romeu just days after his arrival from Chelsea.
The Spaniard soon had a booking but the hosts were now looking more confident, with Jose Fonte heading wide and Pelle denied by a last-ditch challenge and then a save soon after.
That high-tempo start soon petered out but Saints were still having chances, with Howard denying Mane from long range.
They could not reduce the deficit, though, and Everton pulled further ahead six minutes from time, with Barkley curling home at the end of a fine team move after impressively cutting inside.
TWEET OF THE MATCH
"What's Lukaku had for breakfast today?? ..
This is more like it!"
@MichelleHinni Michelle Hinnigan, captain of Everton Ladies
PLAYER RATINGS
Southampton
Maarten Stekelenburg: 6 (out of 10)
Cedric Soares: 6
Maya Yoshida: 5
Jose Fonte: 6
Matt Targett: 6
Steven Davis: 6
Victor Wanyama: 6
Dusan Tadic: 6
Shane Long: 6
Sadio Mane: 7
Graziano Pelle: 6
Subs
Oriol Romeu: 6
Jay Rodriguez: 6
James Ward-Prowse: 6
Everton
Tim Howard: 7
Seamus Coleman: 7
John Stones: 7
Phil Jagielka: 7
Brendon Galloway: 6
James McCarthy: 7
Gareth Barry: 6
Arouna Kone: 7
Ross Barkley: 8
Tom Cleverley: 7
Romelu Lukaku: 8
Subs
Steven Naismith: 6
Tyias Browning: 6
Gerard Deulofeu: 6
STAR PLAYER
Ross Barkley: Ran the show in front of England manager Roy Hodgson
Capitalised on the space afforded to him, setting up Romelu Lukaku's second and curling home a superb goal at the end.
MOMENT OF THE MATCH
Romelu Lukaku: After Tim Howard produced a fine save to deny Sadio Mane, Everton broke at pace and the Belgian headed home a wonderful opener
Lukaku added a second but it was his pre-match actions that proved the moment of the match
Having hit a woman in the stands during the warm-up, he ran up to check she was okay and gave her a hug and a shirt.
VIEW FROM THE BENCH
Saints manager Ronald Koeman - back on the bench after Achilles surgery - again named an attacking line-up, but his side looked unbalanced against Everton
Roberto Martinez's side proved a handful throughout and ruthlessly exposed the space afforded to them in midfield.
MOAN OF THE MATCH
A number of Everton fans let themselves down by singing through Southampton's pre-match tribute to the late Markus Liebherr
The Swiss entrepreneur saved Saints from liquidation and the club remembered their former owner five years after his passing.
WHO'S UP NEXT
Everton v Manchester City (Premier League, Sunday 23 August)
Southampton v FC Midtjylland (Europa League play-offs Thursday 20 August)
Source : PA
Source: PA