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.
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.
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.
