Roberto Martinez's side had their vulnerable defence thoroughly exposed as they took a hammering from the Ukrainian side, who reached the Europa League quarter-finals
Everton were comprehensively dismantled as Dynamo Kiev fought back from a 2-1 first-leg defeat to reach the Europa League quarter-finals, winning 5-2 on the night and 6-4 on aggregate.
Though Andriy Yarmolenko opened the scoring with a phenomenal dipping shot midway through the first half, Romelu Lukaku got one back for Roberto Martinez's side with a similarly impressive effort from the edge of the area.
But Everton's defence was thereafter repeatedly exposed, with Lukasz Teodorczyk and Miguel Veloso scoring a quickfire double to ensure that the tie was firmly in the hosts' favour at half-time.
Things did not look up for the Toffees in the second half, with Oleg Gusev putting another past Tim Howard with his second bite of the apple shortly after the restart. Antunes then compounded Everton's misery with a fine goal from long range.
Phil Jagielka got one back when he headed in a late corner but it was too little, too late for the Merseysiders, whose exit marked the end of Premier League representation in this season's competition.
Gusev and Antunes had early efforts on goal for Dynamo and it was no surprise to see the home side take a 21st-minute lead through the excellent Yarmolenko. The forward was afforded too much time and space and took advantage by cutting inside Gareth Barry before smashing past goalkeeper Tim Howard with his left foot from distance.
His outstanding strike brought the match to life and Everton almost mustered a quick response when Ross Barkley dragged a left-footed strike against the right post, while Yarmolenko was only denied a second by an excellent reaction save from Howard.
Martinez's side regained their aggregate lead just before the half-hour mark, though, as Lukaku pounced on a loose ball, took a touch and curled into the left corner from outside the penalty area for his eighth Europa League goal in five matches.
Dynamo's response was superb, however, as Poland forward Teodorczyk - one of three changes the hosts made from the away leg - scored from close range in the 35th minute. Gusev's ball into the penalty area from the left hit Yarmolenko but Teodorczyk reacted quicker than both Seamus Coleman and Antolin Alcaraz, sweeping home from seven yards.
A shell-shocked Everton then conceded again just two minutes later, as Veloso beat Alcaraz to win a hopeful long ball. Jagielka intervened but could only nod the ball back into the path of the Portuguese, who saw his shot from the edge of the penalty area deflect off the onrushing Coleman, over a helpless Howard, and in.
Still trailing by only one goal on aggregate, Everton sat back at the start of the second half, knowing just how crucial another away goal could be.
But, as a result, they invited Dynamo pressure and Serhiy Sydorchuk took advantage, skipping down the right before cutting back for Gusev who, although his initial effort was blocked by Coleman, reacted swiftly to tuck away the rebound via the right post.
Everton threw bodies forward after falling two behind as Christian Atsu drew an excellent save from Dynamo goalkeeper Oleksandr Shovkovskiy, while Barkley was again denied by the woodwork.
Lukaku and Leon Osman had penalty claims waved away before Antunes provided yet another highlight for the hosts when he hammered a strike into the top-right corner from distance.
And that meant that Jagielka's late header - which came just after Yarmolenko hit the post - was no more than a small source of joy for the frustrated travelling fans, who were left to rue Everton's sloppy defending.
