Maurice Shanahan hit 13 points and fired Waterford to a 0-21 to 1-14 win over Offaly at O'Connor Park Tullamore.
Shanahan was sublime for the Deise men, who recovered from the concession of an early Shane Dooley goal to power to the win with a determined finish, despite hitting 17 wides.
Michael Ryan's side now face Westmeath in the next round, where they will be hot favourites to progress after an improved display.
Midway through the second half it looked as if the home side would push on to the win after they overturned a four-point half-time deficit, but led by Shanahan and the excellent Brian O'Sullivan, Waterford finished the stronger.
Waterford got off to a great start with a couple of quick Jake Dillon points, but they soon found themselves on the hind foot when Shane Dooley punished a poor defensive hand pass by Liam Lawlor and rattled the net for Offaly in the seventh minute.
However the goal was tinged with negativity for the home side as centre-forward Derek Molloy was stretchered off with a nasty look leg injury picked up in the lead up to the score.
After five minutes of a stoppage to deal with Molloy's injury, Offaly did well to hit the ground running again and Dan Currams landed a battling score for the home side after Kevin Brady's arrowed pass picked him out in the corner.
At that stage Offaly looked dominant as David Kenny was managing to keep Shanahan under control, while centre-back Ger Healion was also performing admirably.
However Shanahan soon went in search of possession and he hit form after a couple of quick frees got him off the mark.
From then until the break the Offaly replies were fleeting and a run of four points in a row from Waterford including three Shanahan points from play gave them a 0-13 to 1-6 lead at the break.
At half-time it was questioned whether Waterford had built up a large enough lead after playing with the aid of a stiff breeze, but Deise fans needn’t have worried.
There would have been moments of concern, however, as two rapid fire Brian Carroll points and another from his midfield colleague Kevin Brady drew Offaly level in the 47th minute.
But it was all square for less than a minute as Brian O'Sullivan hit a great score from the wing, and Shanahan and co wrapped up the victory in the closing stages.
Waterford: S O'Keffe; S Fives, L Lawlor, N Connors; J Nagle, M Walsh, D Fives; P Prendergast (0-01), K Moran (0-01); J Dillon (0-02), S Prendergast, S O'Sullivan (0-01); J Barron, M Shanahan (0-13, 0-05f), B O'Sullivan (0-03).
Subs: P Mahony for Barron (35+5 mins), M O'Neill for S Prendergast (58 mins), R Foley for D Fives (63 mins).
Offaly: J Dempsey; C McDonald, D Kenny, D Mooney; R Hanniffy, G Healion, D Morkan (0-01); B Carroll (0-02), K Brady (0-02); C Egan, D Molloy, C Slevin; S Dooley (1-04, 0-04f), D Currams (0-02), J Bergin (0-02).
Subs: C Mahon (0-01) for Molloy (12 mins), P Cleary for Mooney (35+7 mins), S Cleary for Slevin (52 mins), S Gardner for P Cleary (56 mins), S Ryan for Brady (70 mins).
Referee: James McGrath (Westmeath).
Highlights and analysis of all the weekend’s action on The Sunday Game from 22:00 on Sunday 23 June on RTÉ Two and RTÉ.ie (Ireland only).
Expert and fan reaction on Take Your Point from 18:00 on Sunday 23 June on RTÉ Radio 1 and RTÉ.ie (Worldwide).