After 42 goalless minutes Divock Origi and Mamadou Sakho found space in the Everton box to power home quickfire close-range headers, leaving the shell-shocked Toffees 2-0 down at the break.
Any hopes of a comeback evaporated when Ramiro Funes Mori was sent off for a stamp on Origi’s ankle, which also ended the Belgian’s involvement.
His replacement Daniel Sturridge put the Reds 3-0 up just after the hour and also had a partial claim on Philippe Coutinho’s strike 14 minutes from time.
The result leaves Jurgen Klopp’s men in seventh, after both Manchester United and West Ham kept their top-four bids alive with wins.
United moved within a point of fourth-placed Arsenal with a 2-0 success over Crystal Palace at Old Trafford.
Right-back Matteo Darmian saw his cross turned over the line by Palace defender Damien Delaney in the fourth minute and went one better early in the second half when he controlled the ball on his chest and lashed home his first United goal.
The Hammers kept pace by seeing off a compliant Watford 3-1 at Upton Park.
Andy Carroll continued his run of form with the opener in the 11th minute before Mark Noble converted a pair of penalties either side of half-time.
Sebastian Prodl pulled one back for the Hornets but their evening ended in disappointment as Troy Deeney missed a spot-kick and Nordin Amrabat was sent off in injury-time.
