Koeman’s side were desperate to return to winning ways following their frustrating draw with lowly Hull City last time out, but it looked as though Claude Puel’s men would take a share of the spoils at Goodison Park.
However, though Jay Rodriguez and Nathan Redmond had gone close for Southampton, the hosts took a deserved lead through substitute Enner Valencia’s first Everton goal 73 minutes into a scrappy encounter.
And Southampton’s resilience swiftly crumbled, Leighton Baines slotting home from the penalty spot before Romelu Lukaku hammered in a thunderous third late on.
The defeat represents a third straight loss for Southampton, while Koeman can reflect on a job well done over the festive period with Everton having won three of their past five games.
Neither side managed to settle in a scrappy opening in which injuries to Cedric Soares and Dominic Calvert-Lewin disrupted play.
Everton were the first to muster an attempt on goal with Ross Barkley’s rasping strike drawing a smart stop from Fraser Forster.
Forster was called into action again to prevent Lukaku’s effort finding the target as Everton built momentum.
But it was Saints that thought they had the lead against the run of play, only for Rodriguez’s goal to be correctly ruled out for offside, while the forward had also handballed when bundling the ball over the line.
Rodriguez went close again, but – after initially engineering space with some sublime footwork – he blazed over from the edge of the area.
Southampton were almost made to pay for that miss shortly before the break, but Kevin Mirallas was unable to find the top-left corner at the culmination of a swift counter-attack.
Everton started sluggishly after the restart and Koeman wasted little time in making a change, introducing Valencia in place of the ineffective Aaron Lennon.
That change sparked the hosts into life and Lukaku should have done better after he burst clear of Southampton’s defence, only to drag his strike wide.
Puel’s side continued to threaten on the counter, though, and Redmond would have put the visitors ahead if not for a fine save from Joel Robles.
Redmond was made to regret that miss soon after – Valencia marking a fantastic cameo with his maiden Everton goal, tucking home from close-range after Forster had saved Lukaku’s header.
Valencia then played a key part in Everton’s second, drawing a clumsy foul from Maya Yoshida inside Southampton’s area and Baines coolly converted the spot-kick by sending Forster the wrong way.
And there was still time for the Toffees to put further gloss on the scoreline. Baines won the ball back with a sliding tackle and Tom Davies – making his full league debut – picked up the loose ball to slip in Lukaku, who hammered emphatically past Forster with his right foot to wrap up the points.
Key Opta stats:
– Everton are unbeaten in their last 12 home Premier League games against Southampton (W9 D3), conceding just six goals in that time.
– Enner Valencia’s goal was his first in the Premier League since January 2016 (when he netted a brace against Man City for West Ham).
– Leighton Baines became the third defender to score 30 Premier League goals after John Terry (40) and David Unsworth (38).
– The Toffees have scored at least two goals in five of their last six Premier League games, failing to score in the other (0-1 vs Liverpool).
– Southampton have lost three consecutive Premier League games for the first time since December 2014 (a run of four).
– Romelu Lukaku has been involved in 15 Premier League goals this season (11 goals, 4 assists) – at least 10 more than any other Everton player.
– Since his debut for Everton in September 2013, only Sergio Aguero (77) and Harry Kane (59) have scored more Premier League goals than Lukaku (54).