Everton made the short journey to Anfield on the back of a nine-match league and cup run.
Liverpool started with Gerrard for the first time since suffering a hamstring problem, and the influential midfielder was at the heart of his side's high-tempo start as Dalglish's side looked determined to avoid an embarrassing fourth straight league defeat.
It needed a stunning seventh minute save by Tim Howard to deny Gerrard after Luis Suarez's defence-splitting pass.
Everton's United States keeper stood his ground to beat away Gerrard's fierce shot, with Jack Rodwell marking his return with a goal-saving block to keep out Jordan Henderson.
In an entertaining start, South African midfielder Steven Pienaar fired over the bar from a good position after clever work by Leighton Baines.
Eleven red cards have been shown in the past 14 meetings between the neighbours from across Stanley Park.
But referee Phil Dowd did not even have to produce a yellow in the first half despite the frantic pace of the derby.
Andy Carroll and Suarez both spurned further chances before Gerrard's inch-perfect lob earned Liverpool a 34th-minute breakthrough after Martin Kelly was denied by Howard.
Any hopes Everton had of clawing their way back into the derby vanished early in the second half when Gerrard produced another deadly finish.
Suarez did all the hard work, the Uruguay striker cutting into the area and breezing past Sylvain Distin as if he did not exist.
Gerrard took the ball off the former Ajax player's toes and gave Howard no chance with a thumping finish in front of the Kop.
It was one-way traffic. Carroll was unfortunate with a low effort which fizzed just wide before full-back Kelly also went close.
Everton had not lost at Anfield by two clear goals for six years.
But Rodwell's attempts to haul his side back into the end ended with Jose Enrique clearing his low cross before Moyes made a triple substitution.
Gerrard added to Everton's misery when he scored from close range in stoppage time after more good work by Suarez.
-AAP
