Winger Simon Zebo scored one try and set up another for Cian Healy as Ireland soared into a huge lead, with Brian O'Driscoll's try early in the second half putting them 30-3 ahead.
Wales fought back superbly, with Leigh Halfpenny and Toby Faletau leading the charge as the hosts chipped away at the deficit.
But Ireland's defence held strong, getting their Six Nations off to a perfect start and consigning Wales to an eighth consecutive defeat.
Ireland started the match with several relatively new faces in the team, whereas Wales - with the exception of Dan Biggar at fly-half - fielded the side which won the Grand Slam last year.
Yet it was one of those new Irish faces, winger Zebo, who almost decided the match in the visitors' favour inside the first 25 minutes.
Winning just his fourth cap, Zebo began his dream afternoon with a try in the 11th minute as O'Driscoll - in outstanding form despite, or perhaps because of, being relieved of the captaincy - drew the Welsh defence and released the winger to run in.
Jonathan Sexton converted, then added a penalty 10 minutes later - and immediately from the restart Ireland scored a second try.
Biggar was caught taking too long over a clearance in the centre of the park just outside his own 22m, and Rory Best successfully charged him down. The hooker hung on to the ball and made it halfway to the line before spinning it out, but the move looked to have ended when Jamie Heaslip's pass to Zebo went too far back.
Yet Zebo lit up the Millennium Stadium with a truly outstanding piece of skill, flicking his left ankle back to flick the ball forwards, then juggling it twice before hanging on to it and continuing the move.
He was hauled down just before the line but his skill had opened up the Welsh defence and Healy was quickly in to score.
Wales quickly made another mistake to give away another three points, with Mike Phillips caught offside and Sexton landing the penalty from the touchline.
The hosts fought back at the end of the first half, camping in the Irish 22m for almost 10 minutes but coming away with just three points from the boot of Halfpenny.
And that was expunged with the final kick of the half as an Ireland break resulted in a penalty deep in the Welsh 22m for the tackler not rolling away. Sexton once again made no mistake, and the teams went in with Ireland 23-3 ahead.
Bad for Wales became worse within three minutes of the restart, the visitors charging forwards and edging towards the Welsh line where O'Driscoll was on hand to pounce and score his 46th try for his country.
At that point, with Ireland leading 30-3, the match seemed over - but Wales refused to lie down. They proceeded to throw everything at Ireland and put their opponents under huge pressure, which led directly to a first international try for Alex Cuthbert, a yellow card for Best, and a try for the superb Halfpenny.
Wales continued to assault their guests with everything in their arsenal, but Ireland's defence remained dogged to the last with O'Driscoll and winger Craig Gilroy both putting in huge tackles on Halfpenny in full flow that will live long in the memory.
The pressure told on Ireland as Conor Murray was yellow carded stopping Faletau scoring, and their resistance finally gave way when replacement prop Craig Mitchell scored a third Welsh try.
Yet the Welsh effort came too late to make any difference and Ireland hung on for a superb, and unexpectedly dramatic, victory.