The sides had played out a dour goalless draw in Copenhagen and the six previous play-off games across the globe had all finished goalless.
But with tensions at their highest, this return leg was a superb contest and the touch paper was lit by Irish defender Shane Duffy scoring a sixth-minute header.
Ireland's inability to score in the first leg meant their lead was as precarious as could be, however, and two goals inside three minutes put Denmark in firm control.
Cyrus Christie was unfortunate to turn into his own goal, but Eriksen's brilliant strike oozed the kind of class needed to settle these ties.
Martin O'Neill's side had only managed to score three goals once in this campaign, away at Moldova, and the task was beyond them in the 63rd minute as Eriksen curled another beauty past Darren Randolph before drilling home his third from close range to start the party among Denmark's travelling support.
Substitute Nicklas Bendtner then added a fifth from the penalty spot in the last minute as qualification was emphatically sealed for the first time since 2010.
Eriksen was the youngest player in that tournament, but more performances like this will be key as Age Hareide hopes for better than the group-stage exit in South Africa, with Ireland left to lick their wounds at home.