Romelu Lukaku put the Toffees in control at the break with two first-half goals, nodding an Arouna Kone cross home for the opener before doubling his tally on the stroke of half time.

A third arrived with just over five minutes remaining in the match as Ross Barkley curled home to seal the visitors' first win of the campaign.

Here, Sports Mole takes an in-depth look at whether Roberto Martinez's side deserved to run out such convincing winners.

Match statistics

SOUTHAMPTON
Shots: 17
On target: 4
Possession: 54%
Corners: 9
Fouls: 11

EVERTON
Shots: 10
On target: 4
Possession: 46%
Corners: 9
Fouls: 10

Was the result fair?

Southampton may have had more possession and more shots, but they can't have any complaints at being well beaten this afternoon. Everton were the better side and, crucially, showed a clinical edge in front of goal, scoring with three of their four shots on target. The hosts were far short of their best for long spells, and the Toffees' two-goal lead at the break allowed them to be conservative with their play in the second half.

It was actually Ronald Koeman's side who made the brighter start to the match, and just seconds before the opening goal they could have taken the lead, only for Sadio Mane to be denied by Tim Howard. Lukaku's header came against the run of play, but from that point on Southampton never really found their stride again. It seemed to knock the stuffing out of them, and the second just before the interval essentially put the game beyond the hosts.

They did start the second half well and Everton were forced to weather a bit of a storm, but they did it relatively comfortably. Southampton had the chances to make more of a game of it, with Victor Wanyama, Graziano Pelle and Mane all threatening, but the Toffees could have had more as well, with Barkley in particular missing a golden chance. The incisiveness in front of goal was the telling difference today, but Everton were better all over the park and deserve all three points.

Southampton's performance

It may only be two games into the new season, but concerns will be starting to grow in the Southampton camp. Only Chelsea conceded fewer goals than them last season, but already they have shipped five this term. They conceded just 13 home goals throughout the whole of the 2014-15 campaign, but they are nearly a quarter of the way there after just one match now. It was the counter-attack that proved to be their undoing today, with Everton's opening two goals coming on the break.

Koeman would have been relatively content with his side's start to the match, and even when Lukaku opened the scoring he would have felt confident of getting back into the game. However, his side never seemed to fully recover from falling behind, and in the end they put in a very disappointing display. They rallied in the opening 15 minutes or so of the second half, putting Everton under sustained pressure, but even then they were limited to just one or two chances.

It was a far cry from the Southampton we saw in the first half of last season, and their poor form from the tail end of the previous campaign appears to have dragged over to the new one. Koeman will need to think long and hard about what went wrong this afternoon and rectify it quickly if he wants another European push this time around as, on the basis of today's evidence, they could be battling it out in the bottom half unless something changes.

Everton's performance

There is a lot of concern amongst Everton fans at the moment, as demonstrated by a banner being flown over St Mary's before kickoff asking for chairman Bill Kenwright to leave, but this result and performance would have lifted much of that gloom. Their display on the opening day was not exactly encouraging, but all over the park today there were reasons for optimism. Not too many people would have backed them to emerge with all three points - it has been 13 years since they won away at Southampton - but there can be no doubt that they deserved the win today.

Defensively they were very sound, with Howard making a crucial save and the centre-back pairing of John Stones and Phil Jagielka excelling. Barkley enjoyed a good game in midfield, while Lukaku looked unplayable at times in the first half. Martinez slightly altered his usual tactics to cater for being away from home, and it worked brilliantly. They got the ball forward to Lukaku quicker, hit Southampton on the break and played a lot of football in the opposition half, rather than their own.

Their counter-attacking was particularly impressive, and that could well be a tactic Martinez chooses to use a lot away from home this term. They had troubles on their travels last season, but this was a superb away performance at one of the hardest places to go in the league. In addition to the three points and a rousing performance, this will give the Toffees confidence that they can get back into the European race this season.

Sports Mole's man of the match

Romelu Lukaku: There were a number of impressive individual performances from Everton today, with Stones, Jagielka and Barkley all enjoying good games. However, there is really only one choice for this award. Lukaku was at his best in the first half, mixing pace and power to terrorise the Southampton defence. He took his two goals really well and played the role of target man to perfection when required. His influence waned a little after the break, but the damage had already been done by the big Belgian.

Biggest gaffe

Things could have been even better for Everton had Barkley tucked away a fine chance in the first half. The ball found its way to the young midfielder just inside the box, but his tame effort was straight at the keeper when he really should have scored. Tom Cleverley should not be absolved of blame either, having put the rebound wide of the target.

Referee performance

Michael Oliver had a couple of big decisions to make this afternoon, including one early on when Lukaku claimed a penalty for a foul right on the edge of the box. Oliver got his decision to award a free kick spot on, though, and enjoyed a good game in the middle throughout.

What next?

Southampton: The Saints turn their attention to European matters when they face FC Midtjylland in the Europa League playoffs on Thursday. A trip to Watford is next up domestically on Sunday.

Everton: Everton, meanwhile, host Manchester City at Goodison Park on Sunday.