Steven Gerrard moved to level fourth in the all-time goalscorers for this fixture on eight goals with a 21st-minute header from a Luis Suarez corner – and then Daniel Sturridge took over.
Sturridge scored twice in three minutes, making it 2-0 on 33 minutes after latching on to Philippe Coutinho’s through-ball.
Steven Gerrard celebrates his eighth Merseyside derby goal
And the Reds were out of sight soon after when the striker controlled a pass from Kolo Toure and lofted the ball beyond Tim Howard for his 13th goal of the season.
Luis Suarez put further distance between the two sides, but Sturridge spurned the chance of a hat-trick when he fired a second-half penalty over the Everton crossbar.
Elsewhere Ashley Young was on target in Manchester United’s 2-0 win over Cardiff, with the returning Robin van Persie also finding the net, while Adam Lallana scored the final goal as Southampton held 10-man Arsenal to a draw.
Jonjo Shelvey gave Swansea City a much-needed boost, scoring the opener in a 2-0 home win over Fulham.
In the Championship Patrick Bamford, on loan from Chelsea at Derby, continued his goalscoring, scoring his side’s first in a 3-2 win over Yeovil – who had been leading 2-0.
