Andy Carroll, Michail Antonio Help West Ham United F.C. Beat Liverpool F.C.

Andy Carroll and Michail Antonio both scored for a second successive match as West Ham United leapfrogged Liverpool in the Premier League table by beating them 2-0 on Saturday. (Read More in English Premier League)

The pair had scored in the 2-1 victory over Southampton that ended a run of eight games without a win and their efforts against Liverpool at Upton Park ensured that Slaven Bilic's side started 2016 on a high.

While Bilic could draw satisfaction from his side's display -- and in particular the return of Dimitri Payet following a seven-game absence with an ankle injury -- for Liverpool manager Jurgen Klopp there was only frustration that his team were unable to build on successive single-goal victories.

Liverpool have now lost six league games this season -- four since Klopp succeeded the sacked Brendan Rodgers in October.

Klopp had warned that his players would have to show grit in order to emerge from this period with their top-four challenge intact, but their display suggested they were feeling the effects of the festive programme.

Liverpool were disjointed and unconvincing, offering only occasional threats, and while they had opportunities to get back into the game, they lacked the cutting edge to make more of the few chances that came their way.

Bilic's side took the game to their opponents during the first half after surviving an early scare when Jordon Ibe sent a shot across the face of the home goal and narrowly beyond the far post.

That simply served to spark West Ham into life and the pace of Enner Valencia and Antonio immediately began to cause problems for Liverpool before the pair combined to give the home side a 10th-minute lead.

Valencia was given too much space on the right, allowing the West Ham forward to deliver a deep cross to the far post, where Antonio leapt above Liverpool right-back Nathaniel Clyne to head past Simon Mignolet.

Lanzini limps off

Carroll might have doubled the lead moments later when he out-jumped Mamadou Sakho to meet Aaron Cresswell's left-wing cross, before Manuel Lanzini's explosive long-range shot struck Mignolet's left-hand post.

Liverpool's response was unconvincing, with Christian Benteke -- scorer of the winning goal in Liverpool's last two games -- unable to exploit an opportunity after a slip by West Ham centre-back James Collins.

The momentum of the game briefly shifted after a lengthy delay while Lanzini received treatment before hobbling off in the 39th minute with what looked to be a recurrence of the thigh injury that recently sidelined the midfielder for a month.

Liverpool gained some impetus after the stoppage and finished the first half in a positive manner, with Emre Can shooting against the bar after good build-up play from Ibe and Roberto Firmino in first-half injury time.

Five minutes after the restart, Philippe Coutinho's excellent pass set Alberto Moreno free on the left and his low cross found Benteke, only for the striker to side-foot wide.

That miss appeared even more costly when Carroll found the net against his former club five minutes later, rising above Clyne to direct a downward header from Mark Noble's cross into the bottom-right corner.

It might have been worse for Liverpool had Mignolet not produced a superb double save to deny first Carroll and then Antonio's follow-up.

Liverpool upped the pressure as the game wore on, but rarely looked like breaching a West Ham defence well marshalled by Collins.