Liverpool took their time in trying to prise a way through the Magpies defence and a swerving Gerrard strike forced Given into desperately punching the ball to safety.
The pressure built as Newcastle continued to be strangers to the ball, and Gerrard responded to the catcalls in emphatic fashion by breaking the deadlock.
He gave Given little chance of reaching his fiercely struck shot after Lucas Leiva, making his first Premier League start, rolled a free-kick to him.
Newcastle's solitary chance of the first half came when Alan Smith controlled a ball on his thigh and volleyed just wide of the far post with an effort across goal.
The Reds should have been two-up on the stroke of half-time and keeper Given dismissed for Newcastle.
He came storming out of his goal as Liverpool counter-attacked and, as Torres tried to curl a shot around him with the outside of his foot, blocked with his arm.
Torres should have side-footed into an empty net from 25 yards when the ball again broke to him but he hit the post, while referee Alan Wiley took no action against Given.
But the visitors wasted no time after the break to add a second with Sami Hyypia flicking a low corner with his right foot on to Kuyt, who guided in with his shin from point-blank range.
Torres twice went wide as he struggled to make the most of regular opportunities coming his way as the home fans vented their frustrations on Allardyce.
They showed their discontent when Joey Barton came on instead of James Milner.
The fans then cheered when Milner finally came on but the jeers were back with the attack-minded Charles N'Zogbia being taken off to make way.
Liverpool showed little sympathy as Babel played a neat one-two with Gerrard and sidefooted in a shot.
The scoreline could have been worse for Newcastle but for Given as the supporters also made their anger with their team and manager clear at the end.
Newcastle assistant boss Nigel Pearson:
"It was a poor performance from us but I thought they were exceptional and we have to put matters right.
"We have to work hard and stay with each other and it will happen, there is no place to hide and we don't plan to.
"We must put these kind of performances behind us because they are not up to the standard we expect from ourselves."
Liverpool boss Rafael Benitez:
"The performance was almost perfect.
"The team played really well and with a lot confidence and they passed the ball around well and kept a clean sheet.
"Our team has kept the momentum and there are a lot of players performing and it's the kind of display that you want to see."
Newcastle:
Given, Geremi, Beye, Rozehnal, Jose Enrique (Carr 78), Smith, Butt, Emre (Barton 51), N'Zogbia (Milner 59), Viduka, Martins.
Subs Not Used: Harper, Edgar.
Booked:
Butt, Beye, Smith.
Liverpool:
Reina, Finnan, Carragher, Hyypia, Arbeloa, Gerrard (Crouch 80), Lucas, Sissoko, Kewell (Babel 58), Torres, Kuyt (Riise 76).
Subs Not Used: Itandje, Mascherano.
Booked:
Sissoko.
Goals:
Gerrard 28, Kuyt 46, Babel 66.
Att:
52,307
Ref:
Alan Wiley (Staffordshire).
