The Hamburg forward opened the scoring with a third-minute header and then after Everton's Nikica Jelavic had cancelled out Sean St Ledger's 19th-minute equaliser, saw his second-half attempt come back off the post and end up in of the net after ricocheting off unfortunate goalkeeper Shay Given.
But for all Ireland's industry and endeavour, Slaven Bilic's men were good value for a victory which left them top of Group C following the 1-1 draw between Spain and Italy earlier in the day.
The defeat - just the third competitive reverse during Trapattoni's four-year tenure and the first on foreign soil - means the Republic head into Thursday night's showdown with the Spaniards knowing their hopes of qualification could effectively be over if they do not come away with at least a point.
Resilience has been the keyword for the 73-year-old Italian since he accepted the challenge of trying to restore Ireland's fortunes, but while there was no shortage of willingness for the fight, Croatia simply had too much quality, and were ruthless enough to accept the slices of good fortune which came their way.
Republic of Ireland emerged from the tunnel before kick-off to be greeted by a sea of green and a torrent of noise as they embarked upon their first game at the European Championship finals for 24 years. But within three minutes, a sizeable proportion of the hope and expectation which has mounted since qualification via the play-offs in November last year had evaporated.
Croatia started much the better and received their reward when captain Darijo Srna's right-wing cross clipped full-back Stephen Ward and sat up for Mandzukic. The striker had slipped as Srna prepared to deliver, but managed to right himself and head towards the bottom corner.
His effort looked to lack power, but Given, whose fitness had been a major talking point for the previous three weeks, just could not get to the ball in time.
Bilic's men scented victory as Mandzukic and strike-partner Jelavic repeatedly stretched Richard Dunne and St Ledger at the heart of Ireland's defence, the signs were ominous.
Full-backs Srna and Ivan Strinic repeatedly made their way forward to occupy wingers Aiden McGeady and Damien Duff in areas of the pitch where they would rather not have been, and with Luka Modric starting to pull the strings in the middle of the field, Ireland were desperately in need of inspiration.
It arrived, firstly in the shape of striker Kevin Doyle and then St Ledger with 19 minutes gone. The combative Doyle was giving Corluka a torrid time and forced the Tottenham defender to concede one of a series of free-kicks. McGeady's delivery to the far post was firm and true and with Corluka out of position, St Ledger crept in front of him to head the Republic level from close range.
The noise which erupted as the ball hit the back of the net was phenomenal and roared on by the bulk of a crowd of 39,550, Ireland started to make a game of it. Given dispelled any fears of his health with a fine diving save from Ivan Perisic's firmly-struck 22nd-minute shot, and he was grateful to see the same man thump a dipping volley wide nine minutes before the break.
Jelavic puts Croatia ahead
However, the Irish looked to be heading in at the break all square until disaster struck with two minutes of the half remaining. The increasingly influential Modric cut inside the covering Robbie Keane to fire in a long-range shot which was blocked, but in his haste to clear, Ward could only slice the loose ball to the delighted Jelavic, who easily beat Given from close range to restore his side's advantage.
Given's luck deserted him totally within three minutes of the restart when Mandzukic powered a header towards goal from a Perisic cross and saw the ball come back off the post, hit the keeper on the head and fly into the unguarded net.
Trapattoni made a double substitution just eight minutes into the second half when he replaced Doyle and McGeady with Jon Walters and Simon Cox, and the latter sent a long-range effort high over Stipe Pletikosa's crossbar within seconds of his arrival. But with Croatia by now firmly in the driving seat and knowing victory was within their grasp, the Republic found themselves struggling to make it out of their own half.
Dunne headed a 68th-minute Duff free-kick straight at Pletikosa, but Croatia were cruising towards victory without having to expend unnecessary energy. Midfielder Ivan Rakitic curled a 77th-minute shot inches wide and Keith Andrews power a header agonisingly past the post at the death, but the points had been secured long before.