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Philippe Coutinho had not scored since the opening day of the season

Liverpool piled the pressure on Chelsea manager Jose Mourinho as Jurgen Klopp secured a hugely impressive first Premier League win since taking charge at Anfield.

Mourinho was recently given the backing of Chelsea's hierarchy after their poor start to the season - but whether their faith remains after this desperate defeat remains to be seen.

Mourinho: There are fights you cannot win

Chelsea made the perfect start when Ramires headed them into the lead after only three minutes but Liverpool's confidence was not dented and Philippe Coutinho equalised spectacularly in first-half stoppage time.

Coutinho put Liverpool ahead with a shot that deflected off John Terry with 16 minutes left before substitute Christian Benteke wrapped up a victory Klopp's side thoroughly deserved.

All the action from Saturday's football

Listen: I’ll turn Chelsea around if they allow it – Mourinho

Listen to all the goals from Saturday's game

Is this the end for Mourinho?

Mourinho has made it clear he will not resign - while Chelsea's board delivered very public backing to their manager in early October.

Since then, however, Chelsea have barely improved and this comprehensive defeat comes on top of the acrimonious loss at West Ham United last weekend and the Capital One Cup exit against Stoke City on penalties in midweek.

Mourinho laughed in frustration when he thought Lucas should have been given a second yellow card with the score at 1-1

And Mourinho, normally so confident and upbeat, carried the air of a beaten man as he stood hands in pockets on the touchline in the closing seconds with defeat assured and Stamford Bridge emptying rapidly.

Mourinho appeared distracted by opposite number Jurgen Klopp's constant, animated dialogue with fourth official Lee Mason, seeming to question what would happen if he acted in a similar fashion.

As ever, all things rest with owner Roman Abramovich at Chelsea, and while Chelsea's supporters rallied to their manager's cause as his name echoed around Stamford Bridge, this was another lame and uninspired performance from a team delivering relegation form rather than looking up the table.

Now we must wait to see if this loss spells the end of Mourinho's second coming at Chelsea.

Klopp's Liverpool get the message

Jurgen Klopp won his first Premier League game at the third attempt and is unbeaten as Liverpool manager after five matches

This was Jurgen Klopp's first Premier League win as Liverpool manager - and the most compelling evidence so far that the manager who arrived in a blaze of near-hysteria following his sabbatical after leaving Borussia Dortmund is getting his message across.

Liverpool were undaunted by falling behind early on, kept pressing in the style their manager demands and in the end simply outran and outfought Chelsea.

Klopp, as ever, was an animated figure on the touchline, not just encouraging his players but also delivering the most fearful verbal volleys in the direction of Nathaniel Clyne and James Milner, in particular, when they did not do as he demanded.

When he arrived, Klopp announced that he wanted to turn "doubters into believers" at Liverpool - this result will have gone a long way towards achieving that aim.

Chelsea's big players go missing

Lucas (right) completed 73 passes in central midfield, almost more than Chelsea pair John Obi Mikel and Ramires managed between them (76)

Chelsea's title challenge last season was built on the goals of Diego Costa, the midfield inspiration of Cesc Fabregas and the creative brilliance of Eden Hazard.

All three have been conspicuous by their absence this season - and so it was here as Liverpool ensured Chelsea were strictly second best.

Fabregas was only given a role as substitute, Costa was out of sorts and often unable to even bring the ball under control while Hazard was subdued and removed before the hour.

Man of the match - Philippe Coutinho (Liverpool)

Philippe Coutinho scored twice in a game for the first time for Liverpool

He has been a shadow of his usual self this season but his flash of brilliance on half-time changed the game and added a second goal for good measure.

Lost for words? Manager reaction

Chelsea manager Jose Mourinho tells BBC Sport:

"I can't express my feelings in any way at all. You are not just here to interview me. You are also here to see the game. What I see from my players was good until the moment they felt it was impossible to do better.

Klopp applauds 'perfect' Liverpool goals

"The fans are not stupid, I don't need to tell them anything. They know how much myself and the players are trying.

"They know why they are getting bad results. The fight goes on, but sometimes there are fights very impossible to win. You go to a fight with different ammunition. I can play against any team or manager."

Liverpool manager Jurgen Klopp on Mourinho:

"I feel for him. He's a great coach. I don't think anyone in this room doubts he's one of the best in the world.

"Things like this happen. I had a similar situation at Dortmund last year. The good thing was no one in the club was in doubt of my situation. I feel for him of course, but it's work."

The stats you need to know

Jose Mourinho has lost six Premier League matches this season. He's never lost more in a Premier League campaign at Chelsea.

Chelsea lost their first game against Liverpool in seven Premier League meetings.

This is the first time Liverpool have netted more than twice at Stamford Bridge in a league game since December 1989.

Philippe Coutinho ended a run of 710 minutes without a Premier League goal.

This is the first Premier League game since Middlesbrough beat Manchester City 8-1 in 2008 in which a Brazilian has scored for both sides.

James Milner, 29, made his 400th Premier League appearance, becoming the third youngest player to reach the milestone after Wayne Rooney and Gareth Barry.

What's up next?

Both sides head into European competition this week.

Chelsea are at home to Dynamo Kiev in the Champions League on Wednesday in a game they can't afford to lose if they are to advance from Group G.

The following night Liverpool go to Rubin Kazan in the Europa League, looking for their first win after three draws in Group B.