The Reds stormed into the lead after 14 minutes when Leighton Baines turned the ball into his own net while attempting to clear a Luis Suarez strike, and their advantage was doubled after the Uruguayan striker nodded a perfectly weighted Steven Gerrard free-kick past Tim Howard.
But Everton fought back and was level 10 minutes before the interval courtesy of goals from Leon Osman and Steven Naismith.
Liverpool was denied all three points when Suarez had an injury-time goal ruled out in dubious circumstances.
When Z-Cars rang around Goodison Park minutes before kick-off, a ferocious roar brought the stadium to life and the travelling Reds broke into You'll Never Walk Alone only to be roundly booed amid the blur of clapping hands and pumping fists that surrounded them in each of the home sections.
And when Marouane Fellaini towered above Martin Skrtel to feed Nikica Jelavic with a delicate nod and the Croat looked for Naismith at the far post only for Jose Enrique to clear, the noise inside Goodison ratcheted up to a whole new level.
In the 13th minute, however, all that could be heard from the terraces was the sound of Kopites celebrating.
Composed, 18-year-old Suso fed a weighted pass into the path of Enrique, who fired the ball across the face of goal towards Raheem Sterling, but Baines bundled the winger to the floor. However, lurking beyond the far-post was Suarez, who took a touch before firing past Howard to where Baines diverted the ball into his own net.
The first booking of the game came unusually late by Merseyside derby standards. Osman was shown a yellow in the 19th minute. From the resulting free-kick, Gerrard fired a superb curling cross in behind the Everton back-line and, as the Blues' defence dropped deeper and deeper to clear, Suarez kept his eyes fixated on the flying ball and calmly nodded it past the advancing Howard.
But the Blues cut the deficit immediately. Brad Jones rushed out to punch a corner clear but his attempt landed perfectly for Osman, who controlled with his chest before driving the ball through a thicket of Liverpool players inside the penalty area and into the net behind.
In the 29th minute, Sterling was shown a harsh yellow card for a foul on Baines. Minutes later Howard protested for what would have been a cruel red card after another foul from Sterling but referee Andre Marriner saw sense.
Everton peppered the Reds' goal with a combination of deep crosses, aimed mainly towards Fellaini, and balls into the left channel for Jelavic to chase.
The home side had seized momentum and it duly equalized. Fellaini turned free of Andre Wisdom and fired into the middle, where Naismith was arriving right on cue to smash the ball beyond Jones and send Goodison rocking.
And with half-time fast approaching, it was the Blues who looked the more likely to score again as Kevin Mirallas wreaked havoc upon the Liverpool defence and in a vain effort to win a free-kick on the edge of the area, Phil Neville threw himself to the ground and was duly booked.
Brendan Rodgers made two changes at the interval, shifting to three at the back. Nuri Sahin came off for Sebastian Coates and Suso departed to be replaced by Jonjo Shelvey. And after minutes of the restart, Sterling's pace carried him clear of the Everton defence, but his finish was off target.
Neat play from Everton left the Reds' defence in tatters on the hour mark. Seamus Coleman and Jelavic exchanged passes before the full-back whipped the ball into the heart of Jones's penalty area, but no blue shirt was advancing on the six-yard box and Wisdom cleared.
Coleman drove a tantalising cross in between Jones and his defenders, and Jelavic was lethal in his movement but wasteful with his finish, as he sent a blistering header into the Park End from yards out.
Phil Jagielka stretched a leg to deny Gerrard in the 82nd minute after substitute Jordan Henderson won the ball back and squared it to him. Moments later, Suarez skipped past several Everton defenders, but Jagielka was down bravely again to block his effort.
The match ended in controversy, as Suarez found the back of the net from a Coates knock-down and was wrongly ruled out for offside.
Liverpoolfc.com journalists' man of the match: Martin Skrtel.