A second-half strike from Steven Gerrard gave the home side a deserved but slim advantage over their rivals which would have shot them back in front of Manchester United in the title race.
But a battling Everton refused to give in and an 87th minute header from Tim Cahill gave his side a valuable point which edged them ever closer to the European places.
Sticking with an unchanged formation that has seen a run of six games without conceding, David Moyes' Everton started with makeshift striker Cahill as the lone forward with Victor Anichebe having to settle for a place wide on the right.
The move could have paid dividends inside the first minute as Anichebe took a high bouncing ball on his chest and struck it low and hard towards the bottom corner, but keeper Pepe Reina got down well to his left and palmed it off the line.
Liverpool's first chance came in the 12th minute after a Gerrard free-kick hit Leon Osman and rebounded to Dirk Kuyt, but the Dutchman failed to steady himself and his seven-yard effort was blocked by a lunging Leighton Baines.
A quick break from the visitors on 26 minutes saw a neat Steven Pienaar cross-field pass find Anichebe free on the Liverpool touchline and his deep ball to the back post was met by Cahill, whose header looked certain to have broken the deadlock but somehow Reina managed to get his fingertips to the ball to push it past the post.
Liverpool should have taken the lead moments later when a pinpoint 40-yard ball from Sami Hyypia split the Everton defence wide open, but an on-rushing Fernando Torres, with only the keeper to beat, clipped the outside of Tim Howard's post.
The second half started with a Liverpool penalty claim inside the first two minutes following a clumsy tackle from Phil Jagielka into the back of Torres.
It was waved away by referee Howard Webb but replays showed the defender got nowhere near the ball.
The home side had two great chances just before the hour mark when a neat string of passes fed the ball to captain Gerrard on the edge of the box.
First his curling effort brought out the best of Howard in the Everton goal who got down well to his left and his save fell at the feet of Hyypia, who should have scored but a last-gasp tackle from Baines kept the scores level.
The breakthrough came in the 68th minute after Everton failed to pick up Gerrard in the middle of the park.
After a short infield pass from Albert Riera, the Liverpool captain had time to pick his spot and found the bottom corner with a trademark low right-footed drive.
Having not conceded for more than 607 minutes of football, the Toffees were clearly rocked by the goal and it looked like the Reds were going to go on and win comfortably with efforts from Kuyt and Yossi Benayoun putting the visitors on the back foot.
Desperately searching for an equaliser, Moyes swapped Pienaar and Anichebe to alternate wings and the move looked to be a stroke of genius after the visitors won a free-kick outside of the box with only three minutes left to play.
A whipped in cross from Mikel Arteta to the near post had Liverpool in disarray and Cahill stole a yard on his man before heading past a stationary Reina, who didn't stand a chance of getting near the ball.