Steven Gerrard's brace gave Liverpool a valuable lead in their Champions League qualifying first leg - Rafael Benitez's first competitive game as manager.
Gerrard put his side ahead with a stunning first-half strike from 25 yards and doubled the lead with another sweet effort 12 minutes from time.
They were denied more goals, with Graz keeper Andreas Schranz in fine form.
Michael Owen was left as an unused sub by Benitez with speculation mounting over a possible switch to Real Madrid.
The Reds started in lively fashion and Sami Hyypia's bullet header from a Steve Finnan free-kick forced Schranz to brilliantly tip over the bar.
Graz passed the ball around neatly but failed to create any clear-cut chances, Jerzy Dudek comfortably claiming a couple of long crosses pumped into the box.
But Liverpool carried a cutting edge and an inch-perfect Gerrard pass sent Milan Baros through, only for the Czech to be denied by a brilliant Schranz stop.
2 mins: Hyypia header brilliantly saved by Schranz
22 mins: Gerrard's strike puts Liverpool ahead
67 mins: Gerrard's long-range effort is ruled out
78 mins: Gerrard scores again from Cisse's good work
Midway through the half Gerrard's genius gave Liverpool the lead, the England midfielder latching onto a loose ball to lash into the top corner from 25 yards.
Baros and Djibril Cisse then created chances for each other, first the Frenchman narrowly failing to convert a right-wing cross-shot, before Baros headed over from eight yards after Cisse's flick header.
Time after time Liverpool looked like opening up the Graz defence, and Harry Kewell's left-wing cross was only just ahead of the despairing Cisse before Dietmar Hamman's piledriver was well-gathered by Schranz.
The second half brought more of the same from Liverpool, Cisse holding off the last defender before testing Schranz with a well-struck low drive.
Finally Graz created a chance of their own, Roland Kollmann latching onto a left-wing cross only to head back across Dudek's goal and wide of the post.
Normal service was soon resumed as Hyypia headed a corner goalwards only for Schranz to make another flying save.
Gerrard thought he had doubled Liverpool's advantage with another scintillating strike from outside the box, but it was somewhat harshly ruled out for a foul in the area by Cisse.
But the magnificent Gerrard was not to be denied a second, crashing home on the half-volley past a stranded Schranz after more good approach play from the industrious Cisse.
Graz AK: Scheuer, Tokic, Ehmann, Ramusch, Standfest,
Muratovic (Daniel Pirker 81), Skoro (Bazina 62), Amerhauser,
Pogatetz, Aufhauser (Sick 72), Kollmann.
Subs Not Used: Almer, Majstorovic, Plassnegger.
Booked: Tokic, Sick.
Liverpool: Dudek, Finnan (Potter 85), Hyypia, Carragher, Riise,
Kewell, Gerrard (Warnock 80), Hamann, Josemi, Cisse,
Baros (Diao 73).
Subs Not Used: Kirkland, Henchoz, Owen, Sinama Pongolle.
Goals: Gerrard 23, 79.
Ref: Alain Sars (France).