Left-back Antunes added a spectacular fifth goal to seal the Toffees' exit and compound a miserable week for English clubs in European competition.
Below, relive our live text commentary.
Good evening ladies and gentlemen and welcome to our live text commentary as Everton look to restore a bit of national pride by becoming the only English side to progress into the quarter-finals of a European competition this season. Who would've thought it back in August?
With Arsenal, Chelsea and most recently Manchester City bowing out of the Champions League in the second round, and Spurs and Liverpool exiting the Europa League at the last-32 stage, the onus is on Roberto Martinez's side to fly the flag for England and reach the quarters.
But let's not get ahead of ourselves because, like getting the giraffe at a restaurant, progressing remains a tall order for the Toffees. Apologies in advance for that joke.
Everton take a very slender 2-1 lead into this second leg in Ukraine after goals from Steven Naismith and Romelu Lukaku cancelled out Oleg Gusev's opener at Goodison Park last week. Is it enough of a cushion? Time will tell, my friends.
Right, enough messing about, let's get into some team news shall we? Fair enough. I'll have this very soon so stay tuned.
Everton XI: Howard; Coleman, Jagielka, Alcaraz, Baines; Barry, McCarthy, Naismith, Atsu, Barkley, Lukaku
Dynamo Kiev XI: Shovkovskiy; Silva, Antunes, Dragovic, Khacheridi; Veloso, Sydorchuk; Rybalka, Gusev, Yarmolenko; Teodorczyk
Everton subs: Robles, Kone, Garbutt, Stones, Gibson, Besic, Osman
Dynamo Kiev subs: Oleksandr Rybka, Mykyta Burda, Denis Garmash, Vitaliy Buyalsky, Vladyslav Kalitvintsev, Artem Kravets, Dieudonné Mbokani
Okay let's start with Everton and I can confirm that Martinez makes four changes from the side that beat Newcastle 3-0. It's the same back four, but changes aplenty in midfield as cup-tied winger Aaron Lennon drops out for Chelsea loanee Christian Atsu, while Steven Naismith, Ross Barkley and Gareth Barry come in for Arouna Kone, Darron Gibson and Leon Osman respectively. Romelu Lukaku spearheads the attack. Let's go.
On to the hosts now and Serhiy Rebrov, who you may remembe from his time at Spurs, makes three changes to his Dynamo Kiev side from last week's 2-1 defeat. One of those comes in defence as Yevhen Khacheridi replaces right-back Domagoj Vida. Vitaliy Buyalsky drops out of midfield for Serhiy Rybalka, while Lukasz Teodorczyk partners Andriy Yarmolenko up front at the expense of Dieumerci Mbokani.
We don't speak Ukrainian, but we can only assume that former heavyweight champion of the world Vitali Klitschko is in attendence tonight. He's the Mayor of Kiev these days so, you know, it makes sense.
Five minutes left then and let's lodge a prediction shall we? Why not, and considering Dynamo Kiev haven't lost a home game all season, this one is really, really tricky and I'm going 3-1 to the hosts, which would see Everton bow out 4-3 on aggregate.
Incidentally, it's precisely 30 years since Everton qualified for a European quarter-final, when they won the 1985 edition of the UEFA Cup Winners Cup.
Some beautiful scenes inside the Olympic Stadium as hundreds of Dynamo Kiev fans chant and clap in unison. They're up for this! Are you? Such a tricky game for the Toffees tonight. Let's go!
Kickoff: Aaaaaaaaaaaaaaaaaaaaaand we're off in the Ukrainian capital. Everton, playing from right to left, get us underway. Big 90 minutes ahead as the Toffees look to end a 30-year absence from a European quarter-final.
Over the bar! Fast and furious start by the hosts as Oleg Gusev, who scored their potentially key away goal last week, headed Danilo Silva's beautiful cross over Tim Howard's crossbar.
Close! This isn't far away at all as Dynamo Kiev continue their assault, and it's left-back Antunes who drills a shot narrowly wide after Gusev caught Everton with their pants down with a quick throw-in.
Shot! This will settle the visitors down as Ross Barkley drills a shot at goal but, despite fumbling the initial effort, goalkeeper Oleksandr Shovkovskiy gathers at the second attempt. Shovkovskiy looked a bit dodgy last week so that might be an area to exploit for Everton.
Everton well on top, and something better might've come from their last attack but, instead of squaring to the unmarked Leighton Baines who was in an acre of space outside the box, Barkley went for glory and his left-footed shot was blocked before it could reach Dynamo's goal. Christian Atsu has looked promising down their right wing.
We mentioned Oleksandr Shovkovskiy's goalkeeping deficiencies, and it seems like Everton do have a shoot-on-sight policy tonight because they're pinging shots everytime they get within 30 yards of goal.
Substitution: We've seen our first sub, and it's only the match ball! The initial ball has seemingly gone a little flat and the referee has replaced it. We'll keep you up to date as UEFA officials attempt to pump the ball back up #prayforthematchball
Romelu Lukaku was such a nuisance last Thursday, and he has started in the same vein this evening. He's such a big unit and he's holding the ball up brilliantly for his teammates. Andriy Yarmolenko looks like Dynamo's brightest player so far. There's a touch of the Cristiano Ronaldo's about him on that right flank.
Brilliant atmosphere here in Kiev as the thousands of fans cheer on their side, but Everton have been marginally the better team. They've coped really well with the partisan crowd thus far. Let's hope for their sake they continue on for the next 70 minutes.
Goal!!!!!!!!!!!!!!! Dynamo Kiev 1-0 Everton (Yarmolenko)
Well, if this is the goal that takes Dynamo into the quarter-finals then it's a fitting way to do it because it's a fine strike. We mentioned that Yarmolenko was Ronaldo-esque, and Dynamo's number 10 executes a few step-overs to deceive Gareth Barry, before bending a sublime left-footed effort around Tim Howard. Different class.
Let's see how Everton respond then because, as we said, they were marginally the better team prior to that goal. As it stands, though, the Toffees are heading out on away goals, just like Arsenal and Chelsea.
Off the post!!!!!!! So unlucky for Everton as Barkley races on to Steven Naismith's over-the-top ball before drilling a left-footed against the post. Might that prove costly?
Save!!!!!!!! End-to-end game here in Kiev, my friends, as the hosts nearly go two goals up and it's Yarmolenko again as the danger man draws a superb finger-tip save from Howard.
Goal!!!!!!!!! Dynamo 1-1 Everton (Lukaku)
Two goals straight out of the top drawer and Everton are level! What a game. Christian Atsu's fierce volley is blocked, but the ball rebounds to Lukaku, who uses Aleksander Dragovic as a shield to bend the ball around him and goalkeeper Shovkovskiy from outside the box. Two absolutely brilliant goals and, as it stands, Everton are heading through.
Chance! Lukaku is causing so much trouble that he might spend a night in a gulag! He plays through Naismith, whose shot is blocked before it can test Shovkovskiy. Still, Lukaku is having a fine game.
Goal!!!!!!!!!!!!!! Dynamo 2-1 Everton (Lukasz Teodorczyk)
What a game this is and Dynamo are team on their way into the quarters now, as Gusev's cross isn't dealt with by Antonlin Alcaraz, who might've got a push in the back from Teodorczyk, but the Pole then volleys past Howard from close range. We've got a cracker on our hands.
Goal!!!!!!!!! Dynamo 3-1 Everton (Miguel Veloso)
Okay this is getting a bit harsh on Everton now as Martinez's side fall 3-1 behind, and it's Portuguese midfielder Miguel Veloso whose deflected volley deceives Howard and nestles in the top corner. The good news if you're an Everton fan is, well, that goal doesn't really change much because if Everton score again they're going through on away goals.
The bad news is that Dynamo Kiev look dangerous with every attack to be honest. They're still in good shape Everton, but Martinez needs to things out defensively because it's men against boys back there at the minute, with Yarmolenko particularly brilliant.
Four minutes added on: Where on earth did they get number? Is it a minute per goal? Anyway, plenty of time for Everton to nick a goal before the break.
Halftime: Dynamo Kiev 3-1 Everton
There's the whistle. Have we caught our breaths? No? That's understandable because that was a ridiculous first half. Yarmolenko curled the hosts into a 1-0 lead in fine fashion, before Lukaku levelled up with an equally brilliant goal. However, it was one-way traffic then as Teodorczyk and Veloso made it 3-1 on the night and 4-3 to Dynamo overall.
Restart! Aaaaaaaaand we're back underway. Will Dynamo hang on or will Everton score the one goal they need to progress? We'll know the answer to both in just 45 minutes.
Penalty appeal! Heart in mouth moment for the hosts there as Steven Naismith goes under a challenge from Dragovic but the referee tells him to get up - and so he should because it was a blatant dive.
It's a little dodgy from Dynamo, who seem content to invite Everton pressure. Do they know there's only one goal in it? The Ukrainians haven't looked as fluent as they did in the first half, and that'll need to change because, as we said, they don't have that much of a cushion.
Goal!!!!!!!!! Dynamo Kiev 4-1 Everton (Gusev)
Is that all she wrote in this brilliant tie? Gusev, who scored last week, puts Dynamo two goals ahead in the tie as smashes home from close via the post after a defensive mix-up in the Everton defence. They've looked at sixes and sevens back there.
Save! Everton nearly come roaring back as Christian Atsu jigs his way into a goalscoring position before firing at goal, but his fierce drive is palmed away by Shovkovskiy. Fine save and there's still time for Everton, who need two goals in the last half-hour. Not impossible, is it?
Here's something you don't see every day as Everton substitute Leon Osman has a row with the fourth official!
Off the bar!!!!!!! Barkley, who was earlied denied by the post, is now denied by the bar as his fierce drive from outside the box cannons off the woodwork. It's a brilliant effort, and he should do the lotto tonight because he's due a bit of luck.
Close! This would've been game, set and match, as Miguel Veloso, who scored the third, goes in search of his brace but his left-footed drive was always curling away from goal and narrowly missed the post.
Everton subs: Martinez makes a double change as Atsu and Naismith make way for striker Arouna Kone and veteran midfielder Leon Osman as Everton seemingly go 4-4-2.
Everton look to be running out of ideas. Lukaku, who has been brilliant overall, needs better service from the wide areas because he's doing it all himself up there at the moment.
20 minutes left then and, as it stands, Everton need TWO goals in order to make their first European quarter-final in no fewer than 30 years, but that wait looks like being extended at the moment.
Chance!!! Dynamo could, and should, have put the tie to bed here, as Teodorczyk fires straight at Howard in a one-on-one. Really poor.
Is this going to compound a miserable week for English teams in Europe? Man City and Arsenal both failed to reach the quarter-finals of the Champions League earlier in the week, so another defeat for English teams might just hurt the national game even more.
Goal!!!!!!!!!!! Dynamo Kiev 5-1 Everton (Antunes)
Sorry but this is such a fantastic goal from Antunes, who seals Dynamo Kiev's place in the quarter-final with a truly brilliant long-range left-footed effort to leave Everton stunned. We've seen some truly spectacular goals tonight. Excellent.
Off the bar! What a hit from Andriy Yarmolenko, who nearly added insult to injury with a fine strike that bounced off the underside of the crossbar. That's unlucky for him.
Goal!!!!!!!!!! Dynamo Kiev 5-2 Everton (Phil Jagielka)
Now then. Have Everton just thrown themselves a lifeline? Phil Jagielka leaps brilliantly to glance home a Leighton Baines corner and, suddenly, Everton are only two goals adrift and we've got about 10 minutes to play! Game on...kinda.
Dynamo yellow: Dragovic goes into the book for fouling Osman and Everton have a free kick right outside the box. What a chance this is...
Damp squib or damp squid? Either way this was useless from Baines, who floats a free kick right into the wall and then drills the rebound harmlessly at goal. You get the feeling that had to go in if Everton were to progress.
Dynamo sub: Scorer of the fourth goal Gusev makes way for Kalitvintsev for the last few minutes.
Four minutes added on:
Shot! Lukaku's been excellent tonight and he's denied a consolation as the Dynamo stopper palms away his fierce right-footed shot. Good save.
Full time: Dynamo Kiev 5-2 Everton (6-4 on aggregate)
That's all she wrote! Everton bow out to compound a miserable week for English clubs in European competition. We've seen from world-class goals tonight and Dynamo Kiev will take some beating in the tournament this season. Everton were under-par defensively, but taking nothing away from Dynamo, who were ruthless when they needed to be.
Anywho, thanks for joining our live commentary and we'll see you next time. RIP English football in European competition for at least another year.