The improved Wales effort, though, still couldn’t stop Ireland securing the triple crown with victories over all three of the British Isles nations and topping the table with 14 points from three matches.
Wales are bottom of the standings with one point, but interim coach Matt Sherratt, who’s taken over from Warren Gatland, can have few complaints over the performance of his side on Saturday.
Ireland, who lost centre Garry Ringrose to a 20-minute red card in the first half, were under pressure for much of the game, especially in the scrum.
But their defence was excellent and they scored tries through No.8 Jack Conan and fullback Jamie Osborne, before the boot of flyhalf Sam Prendergast carried them to victory.
Wales were vastly improved from the side who slipped to dour defeats under Gatland, and scored tries through flanker Jac Morgan and winger Tom Rogers, before falling short of what would have been a sensational victory.
Ireland took an early lead thanks to a try from Conan, but Wales settled into the game and put pressure on the visitors.
Four scrum penalties and the red card for Ringrose helped their cause and having trailed by 10 points, Wales led 13-10 at halftime.
Ringrose received a yellow card, later upgraded to red on bunker review, for a dangerous tackle on Wales centre Ben Thomas that resulted in head-on-head contact.
Ireland landed a penalty before Bundee Aki came on at the completion of the 20-minute red card and the visitors were restored to 15 players.
They scored a super second try as a cross-kick to the corner was athletically tapped back by winger James Lowe into the hands of Osborne who levelled the game.