Liverpool 3 Olimpija 0
Liverpool win 4-1 on aggregate
Liverpool eased imperiously into the next round of the UEFA Cup with a display which will do as much good for their domestic confidence as their European ambitions.
Gerard Houllier's side needed an opportunity to play with freedom and style against outclassed opponents after successive Barclaycard Premiership defeats, and they took their chance with grateful hands.
Never remotely in any sort of danger, Liverpool needed to confirm in their own minds that their new open, attacking style could work.
And against a side who talked tough beforehand accusing Liverpool of being rich boys without any heart, the Anfield men cruised to victory.
Steven Gerrard, skipper for the first time, was exceptional while teenage French midfielder Anthony Le Tallec scored one goal and had a hand in most of Liverpool's good moments.
Even an El-Hadji Diouf missed penalty could not stop Liverpool repairing the damage from the drawn first leg.
The Slovenians made their intentions very clear, with just one up front and a line of bodies parked 30 yards from their goal. For Liverpool it was the invitation they needed and they battered away relentlessly.
Heskey could have had half a dozen goals in the first half - but in the end had to make do with Liverpool's second.
Young Le Tallec, a bundle of energy and invention throughout, got the first goal on the half hour, but Liverpool should have been out of sight long before the break.
Le Tallec saw a flicked header from Harry Kewell's cross held by Borut Mavric, and the overworked Slovenian goalkeeper had to be equally alert two minutes later when Heskey turned and drilled in a shot which was held low down.
The only early scare for Liverpool came from a badly misplaced pass by Igor Biscan which almost let Olimpija in. Mladen Rudonja raced away and fed Nedim Jusufbegovic, who saw an angled shot turned over by Jerzy Dudek.
Heskey continued to waste chances, mishitting an effort wide before Kewell had a 20-yard free-kick saved and Sami Hyypia climbed to head wide.
The barrage continued with another Kewell shot just wide and then a curling John Arne Riise free-kick which Mavric touched over the bar.
The pressure was constantly on Mavric, and he produced a fine save with his legs after Riise's cross had been hit first time by Kewell from 15 yards.
Liverpool took the lead after half an hour when Mavric punched a Diouf cross straight to Le Tallec, for the youngster to score his first goal for the club.
Heskey went close with two more efforts before he finally got on the scoresheet in the 37th minute.
A fine Liverpool move down the right ended with Smicer carefully laying the ball in for the England striker to finish from six yards and the game was then over as a contest two minutes into the second period.
Le Tallec was again involved, playing a neat pass into Heskey's stride before taking a painful kick for his troubles. He was still on his knees as Heskey lashed over a fine low cross for Kewell to meet on the run and drive home.
The impressive Le Tallec had been subjected to a fearful battering from the Slovenian defenders, and he finally limped away to a great ovation from the home fans on 61 minutes, his countryman Florent Sinama-Pongolle taking over. Next into the action was Henchoz in place of Smicer, with Biscan moving into midfield.
Biscan almost scored with a hooked shot which bounced down and up for Mavric to tip over before Diouf squirmed his way into the box to send Mavric plunging to the other side of the goal to haul down another shot.
Salif Diao replaced Gerrard on 76 minutes, and 60 seconds later Steve Finnan was brought down by Kmetec, although Diouf saw the resulting penalty saved by Mavric.
The margin of victory could have been more convincing, but Liverpool will be content with a display which will go a long way to reviving sagging morale.