Southampton came out on the wrong side of a five-minute blitz at the start of the second half, despite taking the lead through Joe Aribo’s well-taken opener at St Mary’s.
Having fallen behind four minutes after the interval, Everton responded to turn the game on its head through quick-fire goals from Conor Coady and Dwight McNeil, and that was the way it stayed.
Saints might well have been 2-1 up themselves, only for Jordan Pickford to deny Ché Adams with a sprawling save, before an even better stop thwarted substitute Adam Armstrong 20 minutes from time.
Ralph Hasenhüttl made a total of six changes to his starting line-up, as Saints returned to action after two weeks away, including three home debutants.
For Duje Ćaleta-Car, it was a first outing for the club, while Juan Larios and Ainsley Maitland-Niles had previously made substitute appearances on the road.
Elsewhere, Stuart Armstrong, Aribo and Sékou Mara also came into the team, with Mohammed Salisu, Romain Perraud, Ibrahima Diallo, Moussa Djenepo, Mohamed Elyounoussi and Adam Armstrong all dropping out.
Saints were first to threaten when the new-look strike partnership of Adams and Mara, starting together for the first time, combined to good effect.
The Frenchman fed the ball into the path of Saints’ top scorer, who successfully lifted the ball over the advancing Pickford, only to see his chip bounce into the side-netting.
Back came Everton, buoyed by their first win of the season last time out. When Maitland-Niles was booked for bringing down Idrissa Gueye, Demarai Gray sent a vicious free-kick curling just wide of Gavin Bazunu’s left-hand post.
Gray threatened again midway through the first half, whipping in a dangerous low cross that Bazunu had to be sharp to repel in front of the poaching Coady.
The Everton winger looked the most likely to open the scoring in a tepid first half, as he surged towards the penalty area, jinked inside and sent a low shot too close to Bazunu, as Saints struggled to assert themselves.
Reason for optimism among the home crowd arrived when Larios found himself in a crossing position and produced a fine delivery that had to be retrieved at the far post by Kyle Walker-Peters, whose cutback just evaded James Ward-Prowse.
Saints were gradually growing in confidence, with Stuart Armstrong increasingly involved.
The Scot twice attempted to feed low crosses into the near-post area, where Mara was lurking, before Maitland-Niles spread the ball wide for Walker-Peters, whose ball in was met by Armstrong on the volley, but a deflection took the sting out of his strike and Pickford was able to adjust and hold on.
As much as the first period had been a tough watch, the second began at an electrifying pace, including three goals in a five-minute blitz.
It was Saints who edged in front when Adams held the ball up and waited for support from Aribo, who took the ball in his stride, needed a couple of touches to get it out of his feet, and thundered a low shot across Pickford and into the bottom corner from 15 yards.
Saints had no time to take control from their newfound position of leaders in the game, as Everton replied in the blink of an eye.
A foul near the left touchline afforded Gray the opportunity to deliver to the far post, where the towering figure of Amadou Onana headed the ball back across goal for the unmarked Coady to level things up.
Remarkably, despite scoring again two minutes later, Everton nearly fell behind in that time.
Aribo’s pass dissected the defence to pick out Adams, who went eye to eye with Pickford, but the keeper thwarted him with his legs, before Armstrong and Ward-Prowse both saw efforts blocked as they followed the ball in.
Saints’ profligacy was brutally punished within seconds when Alex Iwobi’s cross looked for Onana, prompting cries for a penalty when Walker-Peters prevented him meeting it, but instead the ball ran loose for McNeil, who took a touch and rifled the ball high into the net inside Bazunu’s near post.
Hasenhüttl wasted no time in making a double change, summoning Adam Armstrong and Perraud from the bench, and later Edozie, in what was now a breathless encounter entirely at odds with the first 45 minutes.
Everton twice went close to extending their advantage, with Onana heading wide from a glorious chance, before Gray went through one-on-one and Bazunu showcased his speed off the mark to shut down the angles, as Maupay comically sliced the rebound into the crowd.
Saints’ best opportunity arrived 20 minutes from time when Adams’s driven cross found substitute Armstrong, who took his time before setting himself to shoot and drawing a spectacular diving save from Pickford.
Ward-Prowse lined up a 30-yard free-kick late on, which beat the Everton wall but not the retreating Séamus Coleman, who scampered back into position to ensure the England keeper would not need to add to his impressive collection of saves.
With six added minutes there was still time for Saints to save themselves, but Ćaleta-Car was unable to convert one last Ward-Prowse set-piece, before Adam Armstrong's sharp shot on the turn was straight at Pickford, who gleefully held on – as did his team.