The match started with a bang for Martin O' Neill's charges when a Robbie Brady free kick angled in to a crowded 18-yard box saw Shane Duffy rise highest to power the ball over the 'keeper and into the net in the sixth minute to send the home crowd wild.
In the 17th minute, Denmark fashioned a chance through William Kvist, forcing Darren Randolph to produce a superb save, palming the ball away for a corner which came to nothing. Moments later, the Irish 'keeper denied Pione Sisto from close range after he was played in by the influential Eriksen.
Cyrus Christie produced a great whipped cross from the right flank in for Daryl Murphy not long after that, but the towering striker's inventive flicked effort hit the wrong side of the net. Brady and James McClean linked up superbly on the opposite wing moments later, the Derry man receiving a glorious through ball from his team-mate before rifling a shot across the face of goal, agonisingly wide of the mark.
Denmark would score their first in the 29th minute when Sisto produced a bit of magic down the left-hand side of the box, dinking the ball into the area for Andreas Christensen to prod the ball goal-ward, and Christie couldn't keep it out on the line as it cannoned off the post and in off his foot.
Denmark took the lead two minutes later. Eriksen found himself in space inside the area, producing the goal of the night, smacking the ball in off the crossbar.
The Danes started the second half the brighter, but it was Ireland who carved open the first clear-cut chance when Duffy almost doubled his tally with a header from close range only for Kasper Schmeichel to deny him at close range.
Eriksen pounced on the counter attack not long afterwards, forcing Randolph into another great save before he had to be alert to the second chance, but still Ireland continued to battle, energised by the half-time introductions of Wes Hoolahan and Aiden McGeady - however when Eriksen found himself in space on the edge of the Irish danger zone in the 63rd minute, he essentially ended the contest with a sublimely-struck long-range strike from 25 yards.
Ten minutes later and Eriksen had his hat-trick to make it 4-1 on the night. After some solid work by Sisto on the left flank again, the ball was laid off for Delaney and although Ward stuck his foot out to block the cross, it got stuck under him and popped out for Eriksen to run on to and smash past the helpless Randolph.
Ireland had chances to score in the second half - through Duffy and Shane Long late on - but the Boys in Green were unable to take them, conceding a Nicklas Bendtner penalty at the death to complete the disappointing finale to what was a positive qualifying campaign overall.
Republic of Ireland: Darren Randolph, Cyrus Christie, Ciaran Clark (Shane Long 71), Daryl Murphy, Robbie Brady, James McClean, Jeff Hendrick, Stephen Ward, David Meyler (Wes Hoolahan 46), Shane Duffy, Harry Arter (Aiden McGeady 46)
Substitutes: Keiren Westwood, Colin Doyle, John O' Shea, Paul McShane, Glenn Whelan, Shane Long, Callum O' Dowda, Conor Hourihane, Scott Hogan, Kevin Long
Denmark: Kasper Schmeichel, Simon Kjaer, Andreas Christensen, William Kvist, Thomas Delaney, Nicolai Jorgensen (Nicklas Bendtner 84), Christian Eriksen, Andreas Bjelland, Jense Stryger Larsen (Peter Ankersen 54), Yussuf Poulsen (Andreas Cornelius 70), Pione Sisto
Substitutes: Frederik Ronnow, Jonas Lossi, Mike Jensen, Jannik Vestergaard, Mathias Jorgensen, Jonas Knudsen, Martin Braithwaite, Lukas Lerager, Lasse Schone,