Everton came from behind to win last week's first leg 2-1, but their advantage mattered little as Dynamo tore them apart in Kiev to reach the last eight of Europe's second-tier competition for the first time since the 2010-11 campaign.
Andriy Yarmolenko set the tone for the hosts, opening the scoring with a 25-yard screamer, and although Romelu Lukaku levelled with an excellent effort of his own, goals from Lukasz Teodorczyk and Miguel Veloso had Dynamo in front - both on the night and in the tie - at the break.
Oleh Husyev, scorer of Dinamo's sole goal on Merseyside last week, then added a fourth before Antunes capped a memorable night with a truly stunning strike from 30 yards.
Captain Phil Jagielka headed a second goal for Everton, but it mattered little as their exit ended the involvement of English sides in European club competition this season.
Husyev 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, and his outstanding strike brought the match to life.
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.
Roberto 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.
Husyev'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.
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 Veloso, 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 Husyev, and although his initial effort was blocked by Coleman, he 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 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.
