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.
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).
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.