Substitute Enner Valencia set up a late Everton flurry by scoring his first goal for the club and then winning a penalty as Ronald Koeman’s side maintained their their grip on seventh place.
Southampton were the better side until Valencia was introduced for Aaron Lennon with the game still goalless after an hour.
It turned out to be a masterstroke from the Everton manager as Valencia, on a season-long loan from West Ham, broke the deadlock from close range after 73 minutes before drawing the foul which Leighton Baines capitalised from the penalty spot.
Everton's players celebrate Enner Valencia's opening goal in the victory over Southampton at Goodison Park
Valencia wheels away in celebration after finding the net 17 minutes before the end of the game
The forward struck from close range to put the Toffees in control of their first game in the new year
Leighton Baines punches the air after scoring Everton's second of the match from the penalty spot
Leighton Baines slots home from the penalty spot to put his side two goals up
Romelu Lukaku added a third in the dying moments to ensure Everton claimed all three points in style
MATCH FACTS, RATINGS, TABLE AND MATCH ZONE EVERTON: Robles, Coleman, Funes Mori, Ashley Williams, Baines, Gana, Davies, Lennon (Valencia 61), Barkley (Barry 85), Calvert-Lewin (Mirallas 12), Lukaku. Subs not used: Jagielka, Cleverley, Holgate, Hewelt. Scorers: Valencia 73, Baines, Lukaku Manager: Ronald Koeman SOUTHAMPTON: Forster, Cedric (Stephens 6), Yoshida, Fonte, McQueen, Ward-Prowse, Romeu, Clasie (Long 78), Redmond, Rodriguez, Tadic (Boufal 67). Subs not used: Davis, Reed, Bertrand, Lewis. Booked: Ward-Prowse Manager: Claude Puel. Referee: Kevin Friend (Leicestershire). Attendance: 38,891 If the football's on, it's on Live tables
Fixtures
Scores Premier League
Premier League
Championship
League One
League Two
Scottish Premiership
Scottish Div 1
Scottish Div 2
Scottish Div 3
Ligue 1
Serie A
La Liga
Bundesliga
Enner Valencia's goal in graphic form, courtesy of Sportsmail's excellent Match Zone service. More stats, graphics and analysis HERE
Romelu Lukaku scored a late third as Southampton ran out of steam despite manager Claude Puel making six changes from their defeat against West Brom on New Year’s Eve.
Koeman had marked the start of 2017 by giving 19-year-old Dominic Calvert-Lewin his first senior start, and 18-year-old Tom Davies his second.
For Southampton, a one-game ban for highly-regarded central defender Virgil van Dijk meant a recall for club captain Jose Fonte to partner Maya Yoshida.
Two early enforced changes disrupted the rhythm of the game between two teams hoping to end the season as “best of the rest” outside the top six.
First, Saints full-back Cedric Soares was helped off with a head wound after crashing into the advertising hoardings. Then Calvert-Lewin limped out to be replaced by Kevin Mirallas
No Everton manager had lost to Southampton at Goodison Park since 1997 and Koeman didn’t want to be the first against his former club.
But after Ross Barkley had been denied by Fraser Forster, it was the visitors who looked more coherent with Jay Rodriguez having a goal ruled out after 25 minutes after meeting a James Ward-Prowse cross with his right foot.
Joel Robles then saved from Rodriguez and it needed the introduction of Valencia for Lennon to rouse an unusually quiet Goodison.
Dominic Calvert-Lewin lies injured on the Goodison Park pitch during Everton's goalless draw with Southampton
Jay Rodriguez can't quite turn the ball home for the visiting Saints as the two sides struggled for a breakthrough
Kevin Mirallas cuts a frustrated figure during the Premier League encounter on Monday
Southampton manager Claude Puel (right) and his Everton counterpart Ronaldo Koeman on the touchline
A young Everton fan shouts his support for his team during the Premier League clash on Monday
Teenager Calvert-Lewin was handed a start by Ronald Koeman. Here he challenges Oriol Romeu and Jose Fonte
Southampton's Cedric Soares receives medical treatment early in the game at Goodison Park
Lukaku goes for goal as Southampton substitute Jack Stephens watches on
Calvert-Lewin was forced out of the game in the 12th minute because of an injury
Dusan Tadic takes aim as Ashley Williams tries in vain to get a block in during the first half
Lukaku and Romeu tussle for possession during the Premier League encounter at Goodison Park
Everton goalkeeper Joel Robles looks on despairingly as Jay Rodriguez tries to direct his header goalwards
Ramiro Funes Mori rises above Fonte to head the ball as the two sides struggled to find the opening goal
Tom Davies is fouled by James Ward-Prowse of Southampton, leading to the midfielder being booked
Referee Kevin Friend points to the penalty spot after a foul on Valencia
After Lukaku had dragged a shot wide and Nathan Redmond had been denied by Robles at the other end, Everton took the lead.
Lukaku initiated the move by working hard to release Seamus Coleman down the right. The full-back’s cross was then headed by the Belgian striker against Forster’s legs.
And when the ball rebounded onto Lukaku’s knee, Valencia was on the spot to fire in from a couple of yards.
The emphatic finish broke the Ecuador international’s duck for Everton at the 12th attempt after joining them on loan from West Ham on deadline day.
It was his first club goal of any kind since last January, though he has scored six times for his country in the meantime.
With the deadlock broken, Southampton themselves looked broke. Maya Yoshida ploughed into the back of Valancia after 81 minutes and Leighton Baines, on penalty duties ahead of Lukaku, comfortably sent Forster the wrong way from the spot.
Lukaku then provided the icing on the cake when he hammered a clever pass by Davies into the roof of the net for his 11th goal of the season.