Perhaps it was fitting that names indelibly associated with the World Cup, Geoff Hurst and Martin Peters, were banded around Liverpool on Saturday. West Ham still haven't won at Anfield since 1963, when Alec Douglas-Home occupied 10 Downing Street, the Beatles were at No1 and, more pertinently, the future World Cup winners scored in a 2-1 triumph.

As a half-century of hurt for the Hammers was extended into a 51st year, England were supplied with an unwanted, and rather unnecessary, reminder that their own long wait for global glory is likely to continue beyond next summer. Courtesy of Friday's World Cup draw, Luis Suárez and Uruguay lie in wait for them in São Paulo on 19 June.

If Roy Hodgson was looking for any pointers on keeping him quiet, West Ham offered few. Suárez was irrepressible and, eventually, they were eviscerated. He scored one goal, played a part in two that were debited to visiting defenders and enabled Liverpool to leapfrog Chelsea and Manchester City to go second.

His extraordinary haul at home now includes 10 goals in four games. There is a case for giving him an 11th, with Liverpool's fourth deflecting in off Joey O'Brien's back, albeit after an exquisite display of trickery and a Suárez shot. "He will probably still claim it," said Brendan Rodgers. "It was a wonderful piece of skill."

Suárez's fondness for the spectacular was illustrated against Norwich on Wednesday. His sole goal – officially, anyway – on Saturday was comparatively mundane, but showed the range in his repertoire. He met Glen Johnson's far-post cross with an accurate header. It was yet another example of why Liverpool did not let him leave in the summer. "He gives us a relentless desire," Rodgers said. "When you have that, you don't want to let him go. It hasn't always been smooth sailing but it has been a real privilege to work with him. He is up there with the best players in the world."

He also has a capacity to make inferior players err. The opener was a case in point. Guy Demel was the unwitting scorer but Suárez nevertheless deserved much of the credit. The Uruguayan cushioned Johnson's pass on his knee and unleashed a half-volley. While Jussi Jaaskelainen parried it, the ball bounced into the net off Demel. He has a habit of inducing own goals, with Saturday's two following another from Fulham's Fernando Amorebieta five weeks ago.

"I wouldn't swap him for anyone," Rodgers said. His captain can appear almost as indispensable but the concern for Liverpool is that they may be without Steven Gerrard for a while. He hobbled off. "It was his hamstring," the manager said. "He felt a little tweak. We just need to see how that is."

Before his departure, Gerrard's dead-ball expertise indirectly led to Liverpool's second. His free-kick was met in rather unconvincing fashion by Mamadou Sakho. However, as his miscued volley was bobbling in, James Collins tried to hook it clear. The ball went in via the underside of the bar with Martin Skrtel insisting he had applied the final touch. Whoever the scorer, given West Ham's reputation as set-piece specialists, it was a galling way to concede.

The deficit was then halved when Skrtel, who scored an own goal at Hull last Sunday, repeated the feat, diverting Matt Jarvis's header past Simon Mignolet. "We had 29 attempts and we could have easily doubled the four goals we scored," Rodgers said. Indeed, Raheem Sterling could have had a hat-trick.

Not that West Ham left grateful for Liverpool's profligacy. "Stewart Downing is in hospital having his achilles and calf stitched up from a nasty challenge from [Jon] Flanagan," Sam Allardyce said. The West Ham manager could not claim the moral high ground, though. Kevin Nolan, a boyhood Liverpool fan, was dismissed for a malicious stamp on the back of Jordan Henderson's right leg. "Kevin Nolan shouldn't have lost his cool," Allardyce accepted.

It came just a minute after Suárez scored and an hour after the disparity in striking resources was highlighted by the visiting fans. Forget Luis Suárez, they urged, "we've got Carlton Cole".