Goals from Mario Mandzukic (3), Nikica Jelavic (43) and an unfortunate Shay Given own goal (48) completed a comfortable win for Slaven Bilic's side, who were easy on the eye for much of the night as they dominated possession. They could probably have won by more against an Irish side who lacked the necessary technique to live with their opponents.
Sean St Ledger headed an equaliser for Ireland on 19 minutes, but their performance lacked any real momentum throughout a sodden night in Poznan with basic defensive mistakes contributing to their downfall despite some wonderful support from their vociferous band of travelling fans.
Giovanni Trapattoni's team must avoid defeat against world and European champions Spain - who drew 1-1 with Italy earlier in the day - on Thursday night if they are to avoid elimination at the group stage.
On this evidence, their final match of the section against Italy on June 18 is likely to be their last game of the tournament.
Ireland had more fans than Croatia among the 43,200 crowd for their first match at the finals of a major tournament since the World Cup in 2002.
There was an air of expectancy for their first match at the European Championship since defeat to the Netherlands in 1988, but the Republic could not have made a poorer start as some sluggish defending was penalised by Mandzukic.
The bright Wolfsburg forward nodded into the net from 12 yards out with Aston Villa goalkeeper Given failing to get back across his line to prevent the ball from spinning beyond him.
It put a dampener on a saturated evening, but there would be brief joy as the Irish equalised with their first real chance of the evening as St Ledger rose deep in the opposition box to plant a header into the net from Aiden McGeady's free-kick.
No team had scored more goals from set-pieces in qualifying than Ireland, but it became painfully obvious the longer the night staggered on that there would be no other way to unsettle the slicker team in blue as Ireland seemed to be left marooned deep in their own half.
Croatia won all three group games at Euro 2008 and have made the quarter-finals in two of their previous three appearances at the Championships. Unlike Ireland, they have a pedigree at such blue-chip events, and a world-class playmaker in Luka Modric, who revelled in picking holes in the Irish defence with his speed of thought.
With opportunities obviously going to be at a premium in open play, Ireland needed to be diligent in defence, but were undone by more self-harm when Stephen Ward panicked in trying to clear a loose ball in his box.
The sliced clearance broke kindly to Jelavic, a figure who scored 11 goals in 16 Premier League matches since leaving Rangers for Everton in January. He produced a deft chip to lift the ball over Given as Croatia headed for half-time leading 2-1.
That became 3-1 when the excellent Mandzukic headed goalwards from Ivan Perisic's cross. The ball hit a post before rebounding back off the diving Given's head to drop into the corner of the net. So much for the luck of the Irish.
Ireland had conceded three goals for the first time at a tournament finals. Despite never lacking their typical smattering of spirit, it was difficult to believe they were not already done for.
Robbie Keane appealed in vain for a penalty when he was tackled from behind on the edge of the box by Gordon Schildenfeld, who was fortunate not to present Ireland with a way back into the match on 62 minutes.
Ivan Rakitic almost added a fourth goal for Croatia on 76 minutes when he curled an effort wide while Keith Andrews headed wide from Damien Duff's corner in five minutes of time added on, but the scoreline hardly flattered Croatia who roll on to a match with Italy in their second Group C outing.
Ireland fans were left to console themselves by doing the Poznan in Poznan.