After a quiet start, Steve Staunton's side created the first opening in the 14th minute when Robbie Keane flashed a shot narrowly wide after Simon Davies had lost possession in a dangerous position.
Wales' best chance of breaking the deadlock came in the 30th minute when Craig Bellamy burst clear and looked certain to score until Shay Given raced off his line to narrow the angle and smother the Liverpool star's shot.
At the other end, Damien Duff was sent clear by Keane in the 37th minute, but Wales keeper Danny Coyne was alert enough to snuff out the danger.
Two minutes later Ireland gave his side the lead. The 20-year-old Manchester City midfielder beat Wales' offside trap, rounded Coyne and, despite the tight angle, rolled the ball into the net for his third goal in only four appearances for his country.
The goal was no more than Staunton's outfit deserved after they had edged the first 45 minutes and the second period was a scrappy affair as Wales struggled to get back into the game.
Indeed, the Republic went closest to adding to their lead after the interval. Lee Carsley's deflected 52nd-minute effort from the edge of the area was easily saved by Coyne and, with 14 minutes left, Kevin Kilbane headed into the side-netting from a tight angle.
In the 81st minute substitute Kevin Doyle, who had replaced Ireland on the hour, curled in an effort that beat Coyne but came back off the crossbar and Keane was then unable to get a shot on goal after being brilliantly set up by Duff.
Ryan Giggs curled a late free-kick straight into Given's midriff to end a disappointing afternoon for the Manchester United star and Wales boss John Toshack.
Wales seemed devoid of ideas throughout the 90 minutes, with Giggs and Bellamy off-colour, as the Republic deservedly secured the three points.