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.
Everton's hopes of qualifying for the group stage of the Europa League and guaranteeing their involvement in the competition until at least December rest on their ability to protect a two-goal first leg advantage in Split this evening.
Ronald Koeman lamented his side's inability to pad the 2-0 lead they had established over Hajduk by half time in the first leg but he knows that a disciplined display and the kind of defensive solidity that has been a characteristic of their season so far should see them through.
Once again, the Dutchman won't be able to call on Sandro Ramirez who continues to battle a heel injury and is therefore a doubt for Sunday's trip to Chelsea as well.
Another new signing, Davy Klaassen, has also stayed behind on Merseyside after picking up a knock in Monday evening's 1-1 draw at Manchester City. James McCarthy, a potential deputy for the former Ajax man, is also ruled out, with Koeman not expecting to see him back in contention for a place until after the upcoming international break… that is assuming he isn't thrown back into action prematurely again by Martin O'Neill.
The rest of the squad should remain unchanged, although Jonjoe Kenny joined the first-team on the plane to Croatia offering another back-up option at right back.
How Koeman approaches the game, both in the context of containing tonight's opponents but also with another tough away assignment in the Premier Leaque just three days later, will be interesting. Sandro's absence means that both Dominic Calvert-Lewin and Wayne Rooney could end up playing three times in the space of a week leading the line.
Idrissa Gueye, meanwhile, will likely be called upon to do the same, with the added responsibility of partially covering for Morgan Schneiderlin on Sunday given the Frenchman's suspension and importance in defensive midfield.
It opens an opportunity for Mohamed Basic, perhaps this evening against Split, while the likes of Kevin Mirallas and Ademola Lookman will be hoping to get a look in after being overlooked at the Etihad on Monday. And will Gylfi Sigurdsson be deemed fit enough to make his first start or will he be kept in reserve for Stamford Bridge?
Hajduk, who have lost only one of their last nine home games in European competition, showed in the first leg that they can be accomplished in moving the ball around and are comfortable in possession to a much greater degree than was the case with Ružomberok in the previous round.
In Hamza Barry they have an industrious midfielder who looked to cause the Blues problems at Goodison last week but they will be without the injured Franck Ohandza, himself a deputy for main target man, Márkó Futács. His replacement, Ahmed Said, helped himself to a hat-trick in Hajduk's 4-0 win over Rues this weekend, however.
The key for Everton will be remaining patient and keeping the ball as long as possible to manage the clock, control the tempo and frustrate their hosts in what promises to be an intimidating atmosphere inside Stadion Poljud.
Kick off: 8pm, Thursday 24 August, 2017
Predicted Line-up: Pickford, Holgate, Keane, Williams, Baines, Gueye, Schneiderlin, Davies, Rooney, Mirallas, Calvert-Lewin
The game is being televised live on ITV4 in the UK and on ESPN in the US, Caribbean, Latin America, New Zealand and Australia