Mason Mount's first-half strike powered Chelsea into the top four and condemned Liverpool to another Premier League defeat.
On an evening to forget for the champions, Mount's goal three minutes from half-time proved the difference at Anfield.
Chelsea leapfrogged Everton into the Champions League places following their impressive 1-0 victory – but the result sent the Reds to a record fifth consecutive home defeat and eighth loss of the season.
Jurgen Klopp's out-of-sorts side are seventh in the table, 22 points behind leaders Manchester City.
Tottenham were the beneficiaries of the controversial handball law as they held on to a 1-0 win at Fulham that boosts their top-four hopes.
After Spurs led in the opening period through Tosin Adarabioyo's own goal, Scott Parker's men had the ball in the back of the net when Josh Maja fired home.
But replays showed the ball cannoned into the hand of Mario Lemina from a Davinson Sanchez clearance when he was a matter of yards away.
VAR ruled the goal out, robbing Fulham of what would have been a worthy leveller and potentially having a huge impact on their quest for survival.
It was far from convincing from Jose Mourinho's side, but they held on to register an important win, staying in eighth position but now five points off the top four.
Richarlison continued Everton's Champions League charge in a 1-0 win at West Brom.
The striker's second-half header lifted the Toffees into fifth, one point behind Chelsea, following a third straight win.
Victory ended the plucky Baggies' mini-revival to keep them nine points from safety as they slipped further towards the trapdoor.
Mbaye Diagne's stoppage-time goal was ruled out for offside as Sam Allardyce's side remained second bottom.
