Gylfi Sigurdsson made his full debut and an immediate impact with a long-range shot that effectively ended Hajduk's hopes of turning the tie around
Everton secured their place in the Europa League group stage, Gylfi Sigurdsson's sublime equaliser and Jordan Pickford's penalty save shutting the door on the Croatian side's attempted comeback in the second in Split.
The Icelandic international made his first start since joining the Blues in a club-record £45m deal and he made a crucial intervention just seconds into the second half with a goal from 40-plus yards.
That cancelled out Radosevic's own spectacular strike that had sent Hajduk into the half-time interval with hope at 2-1 on aggregate. But having been dealt the blow of an away goal for Everton when Sigurdsson scored, they saw their hopes effectively die when Ahmed Said was foiled from the spot by the visitors' new goalkeeper later in the second period.
In addition to Sigurdsson's debut, Muhamed Bešić made a long-awaited return to the starting XI after overcoming a long-term injury that ruined his 2016-17 campaign. He lined up in a midfield that is rounded out by Morgan Schneiderlin and Ademola Lookman.
Dominic Calvert-Lewin, scorer of the winner in the away leg in the previous round, started up front with Wayne Rooney playing off him.
Everton had an early scare when Erceg sprung the offside trap and was taken out by Pickford but was flagged offside, incorrectly as it tuner out from TV replays.
The visitors were looking to retain possession and try to patiently wait their chance, the first of which fell to Rooney but he couldn't get enough purchase on Martina's right-wing cross.
Shortly afterwards, Sigurdsson went close with a volley that he sliced a yard wide.
Back at the other end, Pickford was forced into the game's first meaningful save as Hajduk finished the half strongly. The young 'keeper made a fine stop to deny Erceg from the angle and keep it goalless on the night.
Everton were undone in spectacular fashion, though, when Radosevic slammed a 30-yarder past Pickford to make it 1-0 in the 43rd minute.
Aaron Lennon came on at the start of the second half in place of Lookman and the Blues were level within seconds of the restart thanks to another belter of a goal. Sigurdsson spotted Stipica off his line and swing a right boot onto a loose ball that floated over the 'keeper from miles out.
That increased the size of the mountain facing Hajduk, meaning they would need to score three times within the regulation 90 minutes to win the tie.
Rooney could have put the contest to bed in the 47th minute but he couldn;'t steer a header on target while Radosevic despatched another shot from outside the box that arced wide of the post.
Sigurdsson found himself in oceans of space behind the Hajduk defence but his cross for Calvert-Lewin in the centre was cut out, crucially for the home side, by Nizic for a corner.
Hajduk had a vital chance to reduce the arrears when Ashley Williams scythed through his man and the referee awarded a penalty but Pickford was equal to Said's spot kick, diving to his right to push it away Keane who cleared.
A couple of minutes later, Calvert-Lewin was put in one-on-one with Stipica but the 'keeper made an excellent save to prevent him killing the tie before Said rattled a left-footer wide for Hajduk.
With six minutes of the 90 to go, Keane sent Rooney away but he cut back on his right rather than going for it with his left and his shot was blocked by a recovering defender.
Periods of disciplined keep-ball by Koeman's men helped see out the final few minutes, with Everton assured of a place in tomorrow's draw in Nyon.
Full details: ToffeeWeb Match page