Keenly aware that on the road ahead lie heavyweights Italy and defending European champions Spain, both Ireland and Croatia had been clear that they needed a convincing start to the tournament.
But only Croatian coach Slaven Bilic's men delivered - to his palpable delight.
"My players have shown they can cope with the pressure and tension.
"Spain and Italy are favorites in our group, but I still think we are able to beat one of them," said Bilic.
Irish counterpart Giovanni Trapattoni refused to be cowed by the loss.
"We have to believe. Every game in this Euro could be a surprise. We have to believe this," said the Italian.
Ireland's goal scorer Sean St Ledger was dejected at the final whistle.
"It was a poor performance by our standards and the goals we conceded were sloppy," said St Ledger, who hadn't scored all season for his club Leicester City.
"I'm happy I scored but disappointed to have lost this game as we had targeted this one in getting something from it given the two other games that lie ahead.
"Still we showed plenty of spirit towards the end and we can still hope to go through to the last eight."
Just three minutes in, Wolfsburg striker Mandzukic moved into position for a cross from captain Darijo Srna, stumbling briefly before steadying himself to beat Irish goalkeeper Shay Given with a powerful header.
Veteran Given - Ireland's most capped player with 123 - appeared uncomfortable as he was caught off guard, conceding his country's first first half goal in 15 matches.
The 36-year-old Aston Villa keeper had been a doubt because of a knee injury but after being able to train fully on Friday suffered no reaction and was cleared to play in Sunday's match giving Trapattoni a boost.
Croatia's ecstatic fans let off flares to celebrate Mandzukic's goal, and did so again to mark the ensuing strikes despite appeals from the announcer, meaning the team looks set to be the latest to be fined by UEFA for the behaviour of its supporters at Euro 2012.
Ireland, roared on by their fans, battled back and levelled in the 19th minute.
Kevin Doyle was brought down by Vedran Corluka, Aiden McGeady delivered a superb curling free kick which found St Ledger running in at the far post, and he headed it past Croatian keeper Stipe Pletikosa.
Ireland's green, white and orange-clad supporters responded by doing "the Poznan", a backs-to-the-pitch jumping dance made famous by fans of the city's top-flight club Lech.
However, two minutes from the break the Croatians restored their lead.
An unfortunate touch by Stephen Ward put Nikica Jelavic onside, and he beat Given.
Three minutes after the break, Croatia made it 3-1 when Mandzukic fired towards the far corner of the net.
The shot hit the post, but bounced off the lunging Given and into the back of the net.
Ireland and their supporters were furious in the 62nd minute when talismanic captain Robbie Keane - their 31-year-old record goalscorer who like Given is at what is likely to be his last international tournament - was brought down from behind in the penalty area.
But Dutch referee Bjorn Kuipers refused to award a penalty.