A survey by Forbes magazine has just named Manchester United, by a comfortable margin, to be the world’s most valuable football club, but George Gillett and Tom Hicks must believe they pulled off an extraordinary deal to purchase Liverpool for “only” £470m.

This was the second game attended by the US tycoons since they became overlords at Anfield. In the first, Liverpool swept Barcelona out of the Champions League. Here, with no regard to going easy on the young, Arsenal were bulldozed into the ground. The sunshine was bright and the Kop could afford to be discerning, only granting their players the normal level of noisy backing once Liverpool’s third goal went in. Oh, and Peter Crouch played like Marco van Basten. George and Tom: it won’t always be like this.

Whether the “race for third place” has truly the importance to give this result wide significance is debatable, but any match between big clubs is always resonant and one manager was left beaming, the other brooding, by the outcome.

Rafael Benitez almost played stand-up comedian in his press conference and could smile because, having been thrashed here in both cup competitions by Arsenal, he felt his team owed their supporters “something special” and had managed to deliver. Arsene Wenger stewed about what he perceived as a lack of focus from his players, in particular the defenders whose performance he classed as “horrendous.”

It was Arsenal’s worst defeat since Manchester United beat them 6-1 in 2001 and he warned that unless his men get their heads back together, the top four place they appear to be taking for granted could evaporate. PSV Eindhoven might have eliminated Arsenal from Europe but Wenger said Liverpool were “highly favourites” to prevail against the Dutch team in their Champions League quarter-final, the first leg of which is played in Holland on Tuesday night. It was an attempt at humour regarding Crouch but this was one of those days when this lampooned lampost of a footballer turned the joke back on everybody.

He scored, after three minutes, with his right foot, with his head after 34 minutes and with his left-foot nine minutes from time: a “classic” hat-trick comprising three classy finishes.

William Gallas and Kolo Toure have seldom played together because of injuries and they will make a formidable partnership for Arsenal one day, but this was not that day. Gallas looked rusty. Toure paid for the petulance that sometimes creeps into his game, caught out of position while protesting the award of a free kick which, taken sharply by Xabi Alonso, led to Crouch’s second goal. It was an example of the intelligence that Alonso brought to proceedings.

His accomplice in the centre of midfield, Javier Mascherano, was even better, patrolling the area like an airport sniffer dog. Cesc Fabregas and Denilson were hounded as if their pockets were stocked with contraband. Benitez was not getting his English wrong when he used the word “fantastic” to describe the performances of Alonso and Mascherano and, for once, princely young Fabregas was given a lesson by his elders in midfield play.

Arsenal played some nice football on their customary breaks but had no control of the game because of what was happening in central areas, where Jamie Carragher and Daniel Agger also played like thoroughbreds.

Only Emmanuel Adebayor escaped the mediocrity which was besetting his side. The Togolese striker struck the inside of the same post twice, once with a superb low shot after turning Carragher, on the second occasion from a header that Jose Reina got a touch to with a prodigious stretch.

It was Gallas, though, who got Arsenal’s goal, chesting over the line after Toure headed on Fabregas’s corner and Freddie Ljungberg touched the ball past Reina. That made it 3-1 with 18 minutes of the match remaining, and a brief period of sloppiness from Liverpool suggested that Arsenal could even get back into the game, but then Crouch completed his well-earned hat-trick.

Mascherano showed he has skill as well as scrapping qualities by keeping a move going by executing a chip to Jermaine Pennant that came with a high degree of technical difficulty. Pennant crossed and Crouch showed lovely control and awareness to tame the ball, allow Toure to commit himself, and chip over Lehmann.

Crouch’s opening goal, coming so early, set the tone for Liverpool’s dominance. The impressive Alvaro Arbeloa played a one-two with Pennant, who found his colleague with a gorgeous backheel, and Arbeloa’s low, early cross was speared past Lehmann by Crouch, sliding in with Toure and extending a telescopic leg.

Goal number two was the kind that if Crouch could score more of he would be unstoppable. Toure fouled the striker 30 yards out and stood arguing with Steve Bennett while Alonso took a quick free kick to Aurelio, who crossed early. Crouch was arriving and Toure’s absence gave him a clear run at the ball. Rising above Gallas and Abou Diaby, he thundered a header inside Lehmann’s near post.

Arsenal were beaten by another cross when Liverpool moved 3-0 ahead on the hour mark, Agger stealing into space to glance home Aurelio’s free kick. There could have been more goals for Benitez’s men: Crouch went close with an angled shot from the edge of the box after holding off, then turning Toure, and Lehmann made a late stop from Aurelio to avoid the ignominy of 5-1.

Steven Gerrard also shot weakly when teed up by Mark Gonzalez and Liverpool’s victory was the more remarkable for the fact that Captain Fantastic, deservedly substituted, for once was Captain Prosaic.

A justifiably delighted Benitez said: “We have a very important Champions League game against PSV Eindhoven coming up and I have a good selection problem after what Peter Crouch has done.

“But the tactics today were chosen specifically to play against Arsenal. They may well be very different in Eindhoven, we will have to wait and see.”

Wenger said: “Defensively our performance was horrendous. The difference was Crouch.”

Star man: Peter Crouch (Liverpool)

Player ratings: Liverpool: Reina 7, Arbeloa 8, Agger 7, Carragher 7, Aurelio 7, Alonso 8, Mascherano 8 (Riise 82min), Pennant 6, Gerrard 5 (Kuyt 56min, 5), Gonzalez 5 (Zenden 69min), Crouch 8

Arsenal: Lehamnn 6, Eboue 6 (Hoyte 82min), Toure 5, Gallas 6, Clichy 6, Hleb 5, Denilson 6, Fabregas 5, Diaby 6 (Rosicky 65min), Adebayor 7, Baptista 4 (Ljungberg 65min)

Referee: S Bennett

Attendance: 43,958