The sides had played out a numb goalless draw in Copenhagen and the six previous play-off games across the globe had all finished 0-0, with tensions at their highest, but 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.
A rousing rendition of the Irish anthem gave way to a pulsating atmosphere that reached fever pitch when Duffy gave the hosts an early lead.
Nicolai Jorgensen's attempted clearance of Robbie Brady's free-kick went straight up in the air and Duffy was on hand to close in and nod over Kasper Schmeichel from close range.
Despite the goal, Denmark's task remained essentially unchanged - requiring a solitary strike to qualify - and Jens Larsen ought to have equalised almost immediately but his decision to volley, rather than head, Simon Kjaer's excellent cross-field pass was a poor one and he scuffed wide.
Randolph saved sharply from William Kvist and Pione Sisto, but a coiled Ireland carried a threat on the break and Daryl Murphy rippled the side-netting before James McClean drilled wide from an acute angle.
Denmark continued to dominate the possession and their reward came just before the half-hour in fortuitous circumstances as Christie unwittingly bundled in after Andreas Christensen's effort came back off the post.
Though they could rue their luck with that, the second was their own fault, as Stephen Ward was robbed on halfway by Yusuf Poulsen and Denmark sprung forward before Jorgensen squared for Eriksen to smash home via the crossbar from 20 yards.
O'Neill sent Aiden McGeady and Wes Hoolahan on for David Meyler and Harry Arter at half time in the search of a creative spark, but an adventurous change in formation left Ireland vulnerable.
Eriksen had already seen a one-on-one chance to seal it squandered, but the Tottenham star made no mistake from the edge of the area with his left foot to end any flickering Irish hopes.
10 – Christian Eriksen has scored more goals in European 2018 World Cup qualifiers (inc. play-offs) than any other midfielder (10). Force. pic.twitter.com/mvYziNoEpO— OptaJoe (@OptaJoe) November 14, 2017
He then brought up his 11th goal of the campaign and his hat-trick in the 73rd minute, reacting sharpest to Ward's muffed clearance and smashing past a helpless Randolph.
The seal was put on Denmark's famous victory as Bendtner - just seven minutes after replacing Jorgensen - won a penalty from McClean and converted it himself, as Ireland were left to rue their gamble backfiring in painful fashion.