Wales defied the odds to win the title last year but the injury hit visitors, without captain Alun Wyn Jones and a number of other experienced players, were outclassed in Dublin.
Ireland led 10-0 at half-time, with Bundee Aki crossing over for the only try, but its dominance truly told in the second half at Aviva Stadium.
Conway added two more and Garry Ringrose joined the scoring, rendering Taine Basham's late try nothing more than a consolation as Ireland recorded a fifth home Test win in a row against Wales for the first time.
Ireland needed just two minutes to score the first try of the tournament through Aki, who had the simplest of run-ins after being picked out by debutant Mack Hansen.
Johnny Sexton added the extras and, after missing a couple of penalties in quick succession, the Ireland skipper kicked over again to pass the 500-points mark in the Six Nations.
Conway collected an offload from Sexton early in the second half, jinked past a couple of opponents and squeezed over at full stretch, the try allowed to stand after a television match official (TMO) check.
Josh Adams was sent to the sin bin for a reckless challenge on Sexton and more misery was to come for Wales as Conway profited from Josh van de Flier's work to double his try tally.
Ringrose breezed through to add a fourth try for Ireland, which was undone late on when Basham intercepted from Tadhg Beirne and dived down under the posts.
