Gylfi Sigurdsson marked his first Everton start with a stunning 50-yard strike as Ronald Koeman's side reached the Europa League group stage at the expense of Hajduk Split.
Leading 2-0 from the first leg, Everton hopes looked shaky when Josip Radosevic unleashed a wonderful, swerving shot from 35 yards beyond Jordan Pickford.
But Sigurdsson, making just his second Everton appearance following his £45m transfer from Swansea, put even that superb strike in the shade just 14 seconds into the second half.
Picking up a loose ball on the right of the pitch, just a few yards inside the Hajduk half, the Iceland midfielder fired an effort high towards goal and over the frantic reach of the back-pedalling Dante Stipica.
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Sigurdsson's moment of brilliance left Hajduk needing three goals to progress and silenced the hitherto boisterous Stadion Poljud.
The hosts might have got one back when Ashley Williams fouled Ante Erceg in the area and the referee pointed to the spot, but Pickford - another big-money summer addition to the Everton ranks - got down low to his right to keep out Ahmed Said's penalty.
Wayne Rooney, named in the starting XI a day after announcing his international retirement, had a chance to win the match on night for the visitors but goalkeeper Stipica saved.
Everton will learn their group-stage opponents when the draw is made in Monaco on Friday (12:00 GMT).
Man of the match: Sigurdsson steals the show
There cannot be many better ways to start paying off a £45m price tag than that conjured by Sigurdsson just a few moments into the second half.
The £50m valuation Swansea placed on the 27-year-old in July raised eyebrows at the time, but in a few fleeting seconds against Hajduk he demonstrated why he is surely worth every penny, particularly in an inflated market.
His tenacity in closing down his opponent and cutting out an attempted pass was meat and drink midfield play, but what followed was sublime, looking up to see the goalkeeper off his line and then having the creativity, confidence and technique to produce an inch-perfect shot as he fell to the ground.
Up to that point and for the rest of the match Sigurdsson was neat and tidy in midfield - almost peripheral at times - but when it counted he was there with a telling contribution.
It is a sight Everton fans are likely to become accustomed to in the months and years ahead.
"That's ridiculous..." Twitter reacts to Sigurdsson goal
'You need to be really clever to see that possibility'
Everton manager Ronald Koeman: "I saw Gylfi's goal, but I don't think everyone on the bench did. It was incredible. I don't have any other word for it.
"You need to be really clever to see that possibility and, after two or three weeks of the season, it could be the goal of the season.
"The penalty save was key as well because there were still 30 minutes to go at that time. Jordan is showing that we made the right decision to sign him. He is still a young goalkeeper and I have been really impressed by what he's done from day one here.
"It is an important position in the team. There's a big future for Everton and a big future for him."
On resting Idrissa Gueye: "It was too risky to start [him]. He had a scan on his hamstring and everything was fine, but we didn't want to take the risk with Morgan Schneiderlin out of the game on Sunday [through suspension]."
Koeman enjoys another unbeaten start - match stats
- Everton have won nine their past 11 European games (including qualifiers), losing the other two matches.
- For the second consecutive campaign under Ronald Koeman, Everton have not lost any of their opening six games of the season in all competitions (W4 D2).
- Everton have now progressed from each of their past two Europa League qualifying phases (also in 2009-10 under David Moyes).
- Gylfi Sigurdsson made his first start in European club competition since March 2014 (for Spurs v Benfica in the Europa League), and also registered his first goal since April 2013 (also for Spurs in the Europa League, v Basel).
- The Toffees registered more shots on target in the opening 14 seconds of the second half than they did in the entire first half.
- Josip Radosevic's opener ended an Everton run of 313 minutes without conceding in European competition under Ronald Koeman.
- After conceding all four penalties he faced at Sunderland (all in the Premier League), Jordan Pickford saved the first penalty he faced as an Everton player.
Everton will not have long to enjoy their success in Split, with a tough away trip to Premier League champions Chelsea to come on Sunday afternoon (13:30 BST).
- 14RadosevicBooked at 74mins
- 5BarrySubstituted forKovacevicat 85'minutes
- 8VlasicSubstituted forKozuljat 84'minutes
- 22SaidSubstituted forTudorat 74'minutes
- 50ErcegBooked at 89mins
- 2Carbonieri Santa Rosa
- 21BesicBooked at 72mins
- 31LookmanSubstituted forLennonat 45'minutes
- 18G Sigurdsson
- 29Calvert-LewinSubstituted forDaviesat 72'minutes
- Sergei Karasev
- Shots on Target