Statistics wise, Southampton & Everton played similarly. Realistically, Everton didn't want it enough.— Ryan Goodwin (@RyanGUK) December 20, 2014
As well as being tasked with breaking their slump, the Saints also had to contend with injuries to key players, but started with good energy and it was they who saw the game's best early chances.
Having dominated possession in the opening exchanges, the home side perhaps should have gone ahead when Graziano Pelle was presented a free header, but the Italian couldn't direct his effort at goal.
As the game progressed Everton eventually became more involved and soon dominated with a spell of good possession of their own, though a comfortably saved shot from Ross Barkley was all the Toffees really had to show for the opening half hour.
However, just when the first half was threatening to become a thoroughly lifeless affair, a dropping ball in the penalty area gave referee Jon Moss a decision to make when a wild swing of a leg from Phil Jagielka upended Shane Long. It seemed a stone-wall penalty, but the official pointed merely for a corner.
From the resultant set-piece justice was done though, as Saints skipper Jose Fonte pressured Lukaku into an own goal following an excellent delivery from James Ward-Prowse. Moments later Pelle then went close to doubling the lead from close range.
When the game resumed after half time Southampton remained on the front front, looking much more at ease and generally composed thanks to their lead.
Everton were really struggling to put together any telling moves and at the other end only a vital interception from Sylvain Distin kept the home side from doubling their lead as the Frenchman cut out a teasing low cross. Shortly after the hour mark the advantage was increased though as Pelle pounced on a knockdown from Long to finish inside the six-yard box.
The Saints then came close to moving out of sight a few moments later when Fonte was denied his second of the game after a powerful header was parried by Howard.
As the second half wore on Everton moves continued to break down before the likes of Lukaku and Samuel Eto'o could test Fraser Forster in the Southampton goal, while the more Martinez's men pushed the more exposed they were leaving themselves at the back.
When Lukaku's bending power-drive was spectacularly kept out by Forster it became clear it wasn't going to be the Toffees' day.
In the closing stages Yoshida put the Saints three goals to the good with a close range header and the referee's final whistle signalled the end of a near perfect day for Koeman and his players, and a nightmare one for Everton.
Southampton: Forster, Clyne, Fonte, Yoshida, Bertrand, Davis, Alderweireld, Long (Gape - 89'), Ward-Prowse (Targett - 80'), Reed, Pelle
Everton: Howard, Coleman, Jagielka, Distine, Baines, Besic, Barry, Naismith, Eto'o, Barkley, Lukaku