Liverpool are all but in the final qualifying round of the Champions League with next Wednesday's second leg of this tie surely just a stroll for the European champions.
They needed to raise their game considerably after a shock early goal by the Lithuanian champions, Giedrius Barevicius sending the home fans wild.
But first half goals from Djibril Cisse and Jamie Carragher restored some respectability to the scoreline before Steven Gerrard blasted home from the spot for his sixth goal of the tournament to give the Reds a 3-1 away win.
Liverpool can now turn their attention to Friday's third qualifying round draw with some confidence and for an easy passage back into the lucrative group stages.
Peter Crouch made his competitive debut and produced the sort of awkward performance - for the opposition - he had been bought for.
And, but for some wasteful finishing at the end, Liverpool would have had a far greater margin to take back to Anfield for the second leg of this second qualifying round tie.
Benitez, without the injured Djimi Traore and the ill Irish right-back Steve Finnan, still turned out a strong side which had skipper Gerrard in from the start despite reports he had suffered a training injury.
Josemi filled in at right-back while John Arne Riise was on the left of defence with Bolo Zenden taking a wide midfield role.
Milan Baros - again left out to protect his transfer market valuation - sat on the bench set aside for the unused reserves, a solitary figure who surely knows his Liverpool days are nearly over.
Aston Villa will soon intensify their interest but there may be a route to Spain with Deportivo, with Benitez still fancying a swap for defender Jorge Andrade.
That, though, was a sub-plot of a game which Kaunas started like a team who had been playing solidly for four months in their domestic league, sharp and organised and with pace up front.
After a day of thunderstorms and pounding rain, the pitch was slippery as well as bumpy and Liverpool needed to asset their authority early.
Gerrard, Irish youth international Darren Potter and Cisse combined to set up Crouch but the high ball into the box was difficult to control and his shot sailed high over.
Kaunas needed no encouragement but they got it on 21 minutes when the ball was fired in from the right by Arturas Rimkevicius and Barevicius appeared on the near post to bundle the ball past Jose Reina.
But the joy of the 8,300 crowd was short-lived with Liverpool sparked into life and two goals in four minutes.
The first came on 26 minutes when Zenden's cross was knocked down in the box for Cisse to hook home.
On 31 minutes, Liverpool were ahead with Carragher's first goal since January 1999. The ball swirled in from a left corner and Carragher met it firmly to bullet a header past Kurskis.
Potter, clearly getting his chance to shine in the event of Luis Figo changing his mind and shunning Liverpool, was a constant danger.
It was a drive from Gerrard that produced a third goal on 53 minutes. His surge into the box was ended by Mindaugas Baguzis's outstretched leg and Danish referee Knud Fisker pointed to the spot.
Up stepped Gerrard, who made no mistake.
Luis Garcia came into the action in place of Potter, by now some heavy tackles and even flaying fists were beginning to fly, the diving of Tomas Tamosauskas not helping the situation.
Fernando Morientes took over from Crouch but Liverpool were wasteful in the closing stages when more goals were there for the taking.
Zenden, twice, should have scored while Cisse and Alonso missed excellent chances to have made next week's second leg even more academic.
