Right, that's all from us. See you at the weekend?
- Play briefly suspended in first half after crowd trouble
- Keane heads his first goal for Everton from Baines cross
- Gueye then slots home from Rooney through ball
- Return leg in Split next Thursday
Everton head over to Croatia next Thursday (20:00 BST), but before then they play Manchester City on Monday night.
City, Croatia, Chelsea on the Sunday. That's a tough week.
A bad night for last year's losing finalists Ajax, who have been beaten at home by Rosenborg.
This image says it all.
A third goal would have been nice for Everton but a clean sheet was even more valuable.
The Hajduk Split players go over to applaud their noisy visiting fans, who thankfully appeared to behave themselves after that outburst in the first half which led to the game being halted momentarily.
Job well done for Everton.
He's back! Mo Besic is in the book for a textbook late challenge on halfway.
Almost job very well done for Everton, as new record signing Gylfi Sigurdsson looks on...
Starting to repay that £30m transfer fee.
Liam Tyrer: Disappointing 2nd half from Everton - sloppy play all over the pitch. Thankfully Pickford is playing well.
Clueless Roberto: Think we should have gone for more goals. Oh and by the way Pickford is a steal at £30 million!
Everton were very good for the first hour but a bit sloppy since. Still, if they come out of the game with a 2-0 lead they will be very happy indeed. Just five minutes or so to keep the ball and don't give any chances away.
Ooooh that's a chance! Two chances for Hajduk Split as we enter the last 10 minutes. First a cross from the left is almost turned in on goal by Nikola Vlasic, and then Jordan Pickford has to scramble to save a low shot down at his left.
Pickford gives Wayne Rooney a blast, Peter Schmeichel style for giving the ball away cheaply. Hysen Memolla then has a rasping 30-yarder saved by Pickford.
Everton on the ropes a touch. A clean sheet is a must.
Naive from Ademola Lookman as he gives away an unnecessary free-kick in the right-back position, but Jordan Pickford comes to claim it well.
Cuco Martina puts in a cross which Ademola Lookman thumps on target, drawing an excellent save, but the flag is up.
Don't get too excited, Everton fans
Johnny Magrino: Far too much hype around this performance. Frankly, if Everton do anything other than win tonight, it's a horrendous result.
Joshua Edwards: Toffees fans waxing eloquently, but it's Hajduk Split people - a third tier Croatian team. Get a grip.
A first appearance for Everton for Mo Besic since 30 April 2016! He's had some horrible luck with injuries but gets the last 15 minutes in place of Kevin Mirallas.
There aren't many goalkeepers around who can do that. Jordan Pickford claims a catch and in one movement hits a pass on the half volley, maybe 65 yards into the run of Dominic Calvert-Lewin. Sensational distribution.
Rather wild challenge from Hamza Barry catches Ademola Lookman on the thigh, nothing given, and then moments later Barry beats two men and drives a shot wide from 18 yards.
A warning sign for the hosts. They can't get complacent and Wayne Rooney passes on some advice which involves the words 'pass the ball.'
Does anybody want to venture a first-choice Everton XI once the transfer window closes? You'd imagine that Gylfi Sigurdsson will play in behind the striker but where that leaves Wayne Rooney and Davy Klaassen is anyone's guess. Your guess.
Do they still need a number nine? Or is that Sandro's role?
Their midfield has been excellent today.
The Koeman-press. Everton's energy and determination is paying off at the moment.
Benjamin Brute-Moore: The work rate in this side is superb especially in midfield.
Pewak: Hard work gets results - the midfield is full of grafters and neat passers.
Midway through the second half, Everton's focus may well switch to keeping a clean sheet now.
A goal there would have made life very different for Everton in next week's second leg. Here's a change as Dominic Calvert-Lewin comes on up front, Davy Klaassen comes off. Wayne Rooney moves back one into the '10' role.
Plenty of encouragement for Everton fans so far in the link-up between Davy Klaassen and Wayne Rooney and they find each other again, Rooney is in on goal but the flag is up - incorrectly.
He beats the ground in frustration and now Hajduk have their best chance, they work it well to Franck Ohandza and Jordan Pickford has to make a decent save to his left.
If AC Milan go on to lose this tie then... The fallen Italian giants have just bagged their FIFTH goal of the evening against Macedonian side Shkëndija.
Former Liverpool and Sunderland striker Fabio Borini has even got in on the act by netting the fourth, Luca Antonelli putting the Rossoneri 5-0 ahead seconds later.
Elsewhere, Athletic Bilbao have scored three times in six second-half minutes to lead 3-2 at Panathinaikos. Last season's beaten finalists Ajax are 1-0 down against Rosenborg, while Marseille have equalised for 1-1 at Slovenian side Domzale.
Idrissa Gueye is having a fantastic game. He's got a goal but he's also dominating his more natural half of the game. Winning tackles, cutting balls out, finding the easy pass.
The Thursday-Sunday split is not everyone's cup of tea but Manchester United were able to get into the Champions League by winning this trophy last season.
Winning a major trophy would be great for Everton wouldn't it? They've got the squad to compete on two fronts now you'd think.
The curse of the Europa League?
I can't help thinking that if Everton qualify for the group stages they will suffer with their league performances. It could well be a case of running before they can walk.
Good start to the half again from Everton as Kevin Mirallas skins his man on the left but can't find a cross, before Leighton Baines bends over a great ball. Ademola Lookman is the man at the back post but he can't win the header.
The first chance of the second half comes Wayne Rooney's way. Ademola Lookman and Cuco Martina link up well on the right, Martina picks out Rooney and he can only head across goal and wide.
Good movement from the (former?) England man.
It's stick or twist time ahead of the second leg. Is a 2-0 lead enough for Ronald Koeman?
Can Everton put the tie to bed this evening?
Everton Nan: 1st half was like a totally different Everton - keep it up boys.
Richard Widdows: Might be asking for too much but if Everton get 4 or 5 then we can leave a couple of players behind for the away leg.
Morgan Schneiderlin was booked for a rough tackle in the first half and has been replaced at the break, on comes Tom Davies.
Out come the teams....more goals to come?
There are a few interviews taking place on the pitch at Goodison at the break - Shayne Ward answering a few questions - and oops, the sprinklers don't miss do they? Some hasty footwork to get out of the way of the water.
Geoffrey Teale: This Everton team is just starting to gel. The potential here is huge. The movement at times in the last 45 minutes - exciting.
Ed McCosh: Excellent first half for Everton. Some really fluid play at times. Lookman has been exciting to watch. All in all, a pleasing start.
Everton are probably good value for their two-goal advantage. Surely they can put this tie to bed tonight?
A good half for Everton, it was all clicking in midfield and up top too. They could, and maybe should, be further ahead. Let's hope that the crowd trouble which halted the match momentarily is not seen again in the second half.
Robert Smith: Lovely goal that. Great awareness from Rooney and striker’s finish from Gueye!
Sean Cafferkey: Anyone that thinks Wayne Rooney is past his best is 100% wrong. Rooney has still got it.
Rat: This new lad Rooney is special. Great signing.
Good strike! Gareth Barry has left Everton this week but his name continues to be mentioned as Hajduk midfielder Hamza Barry wraps a 30-yarder straight at Jordan Pickford.
Oh and Wayne Rooney should score! Great touch again from Davy Klaassen to play him in, Rooney tries to slide it past Dante Stipica but the keeper saves.
A really good chance.
Jordan Pickford is on his toes to field a shot from Nikola Vlasic which was straight at him.
A really good goal! Davy Klaassen again winning the ball back with some pressing which would be admired from across the park, he then finds Wayne Rooney on the edge of the area and he does brilliantly to hold off a defender and then play Idrissa Gueye in on goal. One touch to set himself, one touch to finish off the base of the post, left-footed. Fine goal.
A line of stewards and police are still surrounding the away end, let's hope that's the end of that. Ademola Lookman does well again to get away from Hysen Memolla, who is booked for pulling him down.