The Toffees continued their impressive start to the season with a commanding win over Nigel Adkins' side, thanks to two goals from their deadly Croatian striker
The Croatian striker returned to the side after shaking off a knee injury, replacing Victor Anichebe in the only change from the team that beat Swansea in the last league match.
Record-signing Gaston Ramirez was given his second start for the Saints, charged with pulling the strings in behind Rickie Lambert and Jay Rodriguez.
However, Ramirez made an immediate impact in a far more decisive manner, rising unmarked at the far post to nod in after Tim Howard flapped at Adam Lallana's sixth minute corner.
John Heitinga almost made amends for his part in the lax marking by getting onto the end of Leighton Baines's dangerous corner, only to head the ball straight into the arms of rookie goalkeeper Pablo Gazzaniga.
Everton slowly began to take control of the match, but were given a reminder of Southampton's threat when Ramirez fired a shot over the crossbar from outside the area.
However, the equaliser arrived with a sense of inevitability for the hosts, as the lively Mirallas fired in a cross from the right that Jelavic failed to control, but fell kindly to Leon Osman, who tucked the ball emphatically into the top corner from all of eight yards.
The goal really lifted the Toffees, and with the wind in their sails they began to apply constant pressure to Southampton's fragile defence.
The visitors have conceded more goals than any other Premier League team this season, and it was easy to see why as Everton burst through the centre to take the lead just after the half-hour mark.
Mirallas was the architect once again, popping up in the centre to play a lovely one-two with Marouane Fellaini before feeding Jelavic, who drove into the left of the Southampton penalty area and pulled the ball crisply back across goal and into the bottom corner.
Nigel Adkins' side seemed rather deflated after falling behind, and succumbed to more and more Everton pressure.
Baines, in particular, was finding acres of space down the left, and his delicious cross was met firmly by Jelavic minutes after his goal, forcing a fantastic save from Gazzaniga.
Unfortunately for the young goalkeeper, that reprieve lasted mere seconds, as his defenders allowed another cross to be played into the box from the resultant throw-in, this time from Seamus Coleman.
Jelavic was lurking unmarked once again, meeting the ball at the far post with a firm header from a tight angle that Gazzaniga could only palm into the roof of the net.
David Moyes's team were really starting to enjoy the game, and would never wanted half-time to arrive as they dominated possession and worked the ball wide at will.
Fellaini seemed keen to get in on the goalscoring act, and had another chance to do so just before half-time, heading Baines's corner onto the crossbar to keep the score at 3-1 heading into the break.
Everton continued to dominate after half-time, with the imperious Fellaini shaking off a couple of challenges to fire in a strike from 25 yards that was well saved once again by Gazzaniga.
Tim Howard was a spectator for most of the second period, but was called into action when Ramirez fired in a stinging effort from 30 yards in the 67th minute, and the Uruguayan came even closer moments later after getting onto the end of a fine Lambert through ball to lift the ball over the keeper but wide of the far post.