Denmark trailed to an early Shane Duffy header but took control immediately and struck twice in three minutes, first when Andreas Christensen's stabbed effort on the half hour mark hit the post and went in off Ireland's Cyrus Christie.
That was followed by a superbly swept home second from Eriksen, who then made sure of victory with a left-footed effort on 64 minutes and completed his hat trick following poor defending 10 minutes later.
Nicklas Bendtner added a late penalty for the Danes, who reached their fifth World Cup having been frustrated by a well-organised Irish display in a scrappy goalless first leg.
"We were well beaten in the end, no question about that... The goals were so sloppy," Ireland manager Martin O'Neill told Irish national broadcaster RTE.
It had all looked so different for Ireland when Duffy took advantage of some poor defending to put the hosts ahead.
Nicolai Jorgensen sliced an innocuous long free kick towards his own goal and into the path of the Brighton and Hove Albion defender who beat Kasper Schmeichel to the ball and sent the packed Aviva Stadium wild.
Denmark refused to panic, however, sewing fears among the home fans that they scored too early as Ireland keeper and first leg man-of-the-match Darren Randolph saved well from William Kvist and Pione Sisto after his defence was carved open.
Ireland tried not to sit back and contributed to the blistering start with two quick chances of their own with big striker Daryl Murphy rippling the side netting with a deft flick and James McClean dragging a shot wide.
Ireland boss O'Neill predicted his side would need to score two goals to end their 16-year World Cup appearance drought and that briefly looked like being the case when Sisto's nutmeg of Harry Arter from a short corner led to Denmark's leveller.
Stephen Ward then carelessly lost the ball on the halfway line and it made its way to Eriksen whose edge-of-the-box strike went in off the crossbar.
The hosts never looked like threatening and fell apart as they were easily picked off on the break with Eriksen curling in the pick of the bunch from outside the box and then hammering home his 11th of the qualification stages.
"It's a crazy feeling right now, we've been fighting together for so long," Eriksen said.
Most Irish fans, in raucous voice only an hour earlier, had streamed out of the stadium before Bendtner completed the humiliation.
(Reporting by Padraic Halpin; Editing by Ken Ferris and Toby Davis)
