Although Ireland gave Croatia a helping hand in all three goals, the 2008 quarter-finalists were deserving winners as they ended their opponents' 14-match unbeaten run and gave them an unhappy start to their first major tournament for 10 years.
Mario Mandzukic, off-balance and unmarked, headed Croatia ahead after two minutes, destroying Ireland's plan to frustrate their opponents.
An unlucky Given own goal at the start of the second half effectively ended Ireland's chances and put Croatia top of the Group C after Spain and Italy drew 1-1 in Gdansk.
"We were much the better team, we were more creative on the pitch," said Croatia coach Slaven Bilic, who will step down after the tournament after six years at the helm.
"From the first minute of the match, I was confident we would be the better team. We showed tonight how we can play."
Ireland coach Giovanni Trapattoni, who at 73 is Bilic's senior by 30 years and became the oldest coach at a European Championship, was non-plussed by his team's defence, which is usually their strong point.
"It was not like our other performances, in many games we have played better, we didn't concede goals, tonight we conceded three goals," said the Italian, whose side's 14-match unbeaten run included 11 clean sheets.
Trapattoni had said before the game that the most meticulous plans can be undone by unforeseen slip-ups but even he was shocked by his team's start.
"Maybe it was the tension," he said.
Mandzukic, who had stumbled from his knees and was unmarked, got up just in time to head Darijo Srna's cross from the right past a stranded Given after three minutes
Although the game briefly became bogged down in midfield, Croatia came into their own as Bilic's bold formation paid off.
Captain Darijo Srna, usually a midfielder, was moved to right back and, with Ivan Rakitic and Ivan Perisic marauding down the wings and playmaker Luka Modric coming to life, their quick and incisive attacks were a pleasant variation on Spain's tiki-taki.
Modric's low pass hit two Irish defenders on its way through the defence and landed at the feet of in-form striker Jelavic who diverted it past Given.
The entire Irish side protested about an offside in the build-up, Trappatoni saying a player had been "two metres offside."
