Tomasz Radzinski had put Everton ahead early in the second half of this enthralling Mersey derby, only for the Frenchman to force a draw.
Everton, who also had Thomas Gravesen back in the fold and on the bench for the first time since his Christmas outbursts, gave Gary Naysmith the job of trailing Danny Murphy, whose dangerous runs from right wing to central midfield can cause plenty of unsuspecting damage.
John Arne Riise was getting similar treatment on the left, and two collisions with David Weir prompted an exchange between the Scot and the Norwegian.
Former Everton defender Abel Xavier was booed, not surprisingly, by the away fans every time he touched the ball as the visitors started confidently. They allowed Liverpool the ball in their own half, but one rapid break saw Stephane Henchoz send Anelka away with a 40-yard pass, the Frenchman racing down the right before drilling a shot into the side netting.
Everton created a good opening on 12 minutes when Scot Gemmill and then Kevin Campbell worked really hard to win possession and a flowing ended with a pass to Carsley, who took on Sami Hyypia and fired in a right-footer that Kirkland clutched to his chest.
Three minutes later Murphy and McAllister worked the ball out to Riise in acres of space and the flame-haired flank man cut inside young Clarke before curling a left-foot shot just under the bar, to be clutched by Simonsen.
Weir, who had been having an outstanding game, was beaten by a clever Anelka flick as he chased a Riise pass and the Frenchman sprinted away before trying to chip Simonsen, but he did not get enough pace on the ball and the danger was cleared from the six-yard box by Naysmith.
Everton could be more than satisfied with the way things were going. Their tactics were frustrating Liverpool and Carsley, Linderoth and Gemmill were winning plenty of possession in midfield.
Everton produced a good move on 24 minutes when Ginola's pass sent Naysmith into the box for a shot that was deflected for a corner, Alan Stubbs heading the flag kick wide.
Liverpool were making mistakes in possession and Hyypia hauled down Campbell to give Ginola the chance to curl the ball in from 25 yards and cause considerable trouble for Liverpool to clear.
A foul by Weir on Owen resulted in a bout of finger-pointing and chest-pushing by the pair, Weir suggesting that the England striker had gone down a touch too easily. Referee David Elleray pulled both aside and made it clear the book would be out next time.
Four minutes from the break, after Stubbs had tripped Dietmar Hamann on the edge of the box, Riise - from a touched free-kick - blasted a shot into the wall.
Clarke was then booked on 43 minutes for sending Riise flying after the Norwegian had touched the ball past him.
In first half injury time, Riise's hook from the edge of the box bounced up and was clawed away by Simonsen, who then bravely threw himself at Hyypia's feet to clutch the lose ball.
HT Liverpool 0 Everton 0
Before the restart, Everton dispensed with Ginola and sent on Radzinski up front and injected Gravesen's battling qualities in place of Linderoth.
Everton's confidence was growing and Campbell had a header blocked, with the ball being cleared frantically away from Stubbs.
Hamann, at the other end, then saw a fierce low drive, blocked by Gemmill after a corner had been only partially cleared.
But from the next break, Everton scored. Alessandro Pistone - playing his first game since December - played a neat one-two with Gemmill before surging down the left.
When the low cross came in, Radzinski scuffed a shot from the edge of the box that bounced away from Kirkland and into the bottom corner.
Heskey immediately came on for Stephen Wright, as Everton's noisy supporters celebrated almost as wildly as their bench.
And on 57 minutes, Heskey created Liverpool's best opening so far with a chip into the box that Owen spun and hooked over as he fell.
Anxiety, apart from a small corner of celebrating Blues, was all around Anfield now. Their side had looked strangely lethargic, confronted with Everton's sheer desire.
Another Owen shot crashed into an Everton body and was hacked away, and only now were Liverpool playing with any pace.
Owen then saw another effort go begging as Heskey chipped in from the left and his striking colleague ducked to head the ball over the bar.
Vladimir Smicer was next into the fray in place of McAllister in an attempt to salvage something for the Reds.
Finally Liverpool's pressure paid off on 71 minutes. Heskey fed Murphy, whose neat pass sent Anelka clear in the area to slide his shot past an exposed Simonsen.
It was all going one way now as Everton dug in to get some sort of reward for an outstanding contribution to a game in which many felt they were only there to make up the numbers.
Owen then surged from 40 yards out before unleashing a shot that Pistone charged down.
From the corner Simonsen produced the save of the match to touch over a Murphy header.
Kirkland almost matched that with a flying one-hander to keep out a Radzinski drive with five minutes left.
Gravesen, who got himself in a few scrapes, was finally booked for tripping Liverpool's man of the match Murphy with a minute to go.