The club's legendary former player and boss went into the game on the back of successive defeats to Manchester United and Blackpool but the result against their closest rivals at least ensured his first point.
However, there are still many issues to address as Liverpool's defensive frailties let an early advantage slip in a frenetic six minutes after half-time and their pride was only salvaged by a penalty in the final quarter.
Raul Meireles picked the perfect time to score his first goal to give the club he joined in the summer an interval lead but Sylvain Distin and Jermaine Beckford soon turned that around after the break.
But the Reds earned a deserved draw when Dirk Kuyt stroked home his fifth derby goal from the spot.
Watched by principal owner John Henry and chairman Tom Werner, Dalglish received the expected rousing welcome which the Kop had been waiting to give him for a week since taking over from Roy Hodgson.
The occasion may have got to his players more than him as Everton enjoyed the better of the early exchanges.
However, it was not long before Fernando Torres showed he was in the kind of mood which had brought 63 goals in his previous 99 Barclays Premier League appearances.
Having volleyed over from a Meireles corner the Spaniard almost opened the scoring in the 16th minute with a typical piece of brilliance.
Anticipating the bounce of the ball better than Distin the 26-year-old cleverly turned the centre-half before twisting inside his defensive partner John Heitinga and unleashing a left-footed shot which beat Tim Howard but rebounded off the far post.
Kuyt charged into the area to reach the loose ball but his effort was deflected over.
Dalglish's decision to play Glen Johnson out of position at left-back in the last two matches had raised a few eyebrows but the England international's right-footedness was pivotal in setting up the opening goal.
Having cut inside in the 29th minute his far-post cross was headed down by Kuyt but Howard saved brilliantly low to his right and got up immediately to block the Dutchman's follow-up.
The Everton goalkeeper may have felt his work was done but the ball dropped to Meireles on the edge of the area and he drilled a swerving shot inside the near post with Howard unsighted.
Meireles was having one of his best games since arriving in the summer and when Kuyt's inch-perfect through-ball picked him out Howard was forced to tip a shot around his near post.
Half-time brought not only a change of player - Sotirios Kyrgiakos replacing the ill Daniel Agger - but a major shift in momentum.
Just 42 seconds after the restart Martin Skrtel allowed Distin to head Mikel Arteta's corner past Jose Reina and Johnson could not keep the ball out.
Everton went ahead six minutes later while Martin Kelly was laying pole-axed on the pitch after landing awkwardly after an aerial collision with Victor Anichebe.
The ball found its way to Leon Osman and he jinked his way into the penalty area before slipping a pass inside for Beckford to slot past Reina, who was booked for his protestations about Kelly not being able to receive treatment.
But just as Everton looked to have secured a grip on the game Howard handed their rivals a way back in the 68th minute when he tripped Maxi Rodriguez as they challenged for the breakdown of Meireles' free-kick.
Referee Phil Dowd pointed to the spot and Kuyt duly dispatched.
The closing moments became increasingly frenetic as the home side pressed for the winner they desperately craved to get the second era of Dalglish up and running.
Even the legendary manager's encouragement from the sidelines could not cajole the breakthrough.
But with Henry and Werner in the city for the next few days he may have to save his most persuasive arguments for the owners as he seeks reinforcements for his squad.
Liverpool are only four points above the relegation zone and considerable improvement is still required to banish the lingering but almost unthinkable spectre of relegation.
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