Ireland's evening started on a high note as Brighton & Hove Albion's Shane Duffy nodded Martin O'Neill's side in front with only seven minutes on the clock, getting his head to Robbie Brady's long free kick to guide the ball over Kasper Schmeichel in the Danish goal.
There were chances at both ends before the Danes drew level under fortuitous circumstances in the 29th minute, Andreas Christensen's close-range effort rebounding off the post and ending up in the net via a wicked deflection off Middlesbrough's Cyrus Christie.
The hosts were reeling after conceding an away goal and they came undone again moments later as Eriksen helped himself to his second, the Spurs midfielder collecting the ball from Nicolai Jorgensen following a Stephen Ward error and smashing it home from close range via the underside of the crossbar.
Ireland failed to close down Eriksen all evening and he was on hand to punish them again just after the hour mark when Pione Sisto picked him out on the edge of the box, and he curled Denmark's third into the top corner.
The Spurs star completed his treble in the 74th minute thanks to another howler from Ward, whose poor clearance sent Eriksen clean through on goal, allowing him to thunder an unstoppable strike beyond goalkeeper Darren Randolph to compound Ireland's misery.
Things would get even worse for the Irish before the evening was over as Nicklas Bendtner went down in the box in the final minute and the referee pointed to the spot, allowing the former Arsenal man to step up himself and put the ball beyond Randolph from 12 yards.
The result means that England will be the only home nation at the World Cup in Russia next summer, with Northern Ireland also dropping out in the playoff phase of qualifying.