Galway withstood a brave second half rally from Waterford to book their place in the semi-finals at Croke Park yesterday.
The beaten Munster finalists got off the mark first with a point from Eamonn Murphy, but Galway scored four in response by the fifth minute.
Neil Keary slotted the first brace, and Shane Moloney followed up with a brilliant score, before Trevor Moran extended Galway's advantage.
Waterford gradually settled into the game, though, with Paudric Mahony converting two frees, and when Murphy fired over following a splendid turn, they trailed by one.
Jonathan Glynn's goal after 15 minutes gave the winners impetus, however, and they streaked clear in the second quarter to lead 1-9 to 0-5.
Points from Conor Cooney and Jason Flynn meant that all six of Galway's forward division had scored from play by the 24th minute.
Moloney and Cooney added scores either side of a point from Waterford substitute Martin F O'Neill.
Signs were ominous for the Decies and they could have done without the unusual decision of referee Garrett Duffy to signal half-time with the Suirsiders preparing to take a '65.'
They got the perfect start to the second half, however, when Jake Dillon's shot for a point in the 32nd minute dipped and was fumbled by Galway goalkeeper Shane Kelly into his net.
Points from Gavin O'Brien and Mahony added to their hopes, but Galway responded through Albert Mullins and Moloney, while Darragh Fives denied Glynn a second goal.
Dillon then pounced for a second goal, finishing smartly in the 45th minute, but Murphy missed an opportunity to level the game and Waterford paid the price as Galway struck four points in as many minutes to open a five-point gap.
Moloney converted three frees in this period, while Padraic Brehony flighted over a decent effort. Though Stephen Roche did get one back for Waterford, it was too little too late.
Scorers -- Galway: S Moloney 0-6(0-4f), J Glynn 1-0, N Keary, C Cooney, A Mullins 0-2 each, P Brehony, J Flynn, T Moran 0-1 each. Waterford: J Dillon 2-0, P Mahony 0-3f, E Murphy 0-2, G O'Brien, S Roche, MF O'Neill 0-1 each.
Galway -- S Kelly; C Diviney, Darragh Burke, C McDonagh; P Hoban, Daithí Burke, M Horan; P Brehony, C Cooney; A Mullins, J Glynn, J Flynn; N Keary, T Moran, S Moloney. Subs: G O'Donoghue for Darragh Burke (ht); P Concannon for McDonagh (57); M Keating for Brehony (60+2).
Waterford -- D Duggan; J Barron, S O'Keeffe, L Egan; R Barry, D Fives, E Madigan; G O'Brien, P Prendergast; P Mahony, E Murphy, C Curran; S Roche, C Heffernan, J Dillon. Subs: MF O'Neill for Heffernan inj (22); H Barnes for Madigan (ht); C Chester for Curran (42); T Curran for Barry (46).
Ref -- G Duffy (Antrim).