An inspired Luis Suarez put the Reds in the driver's seat, but Brendan Rodgers' side threw away a two-goal cushion as David Moyes' men responded well to claim a deserved point.
Two early Liverpool goals - with Luis Suarez at the heart of them - left David Moyes' team shell-shocked, but the Toffees responded superbly with Leon Osman and Steven Naismith cancelling out a Leighton Baines own goal and Suarez's header.
Marouane Fellaini returned for the home side after recovering from a knee injury, while Kevin Mirallas took the place of the suspended Steven Pienaar on the wing.
Brad Jones continued in goal for Liverpool despite Pepe Reina being fit enough for the bench, with Jose Enrique starting as the injured Glen Johnson missed out.
After a slow start the game sprang into life after 13 minutes with the Reds getting a fortuitous early lead.
Raheem Sterling appeared to be pushed over by Baines as he shaped to finish an Enrique cross, but referee Andre Marriner waved play on, and Suarez's fierce strike across goal hit the England left back and ricocheted into the net.
The Uruguayan forward ran the risk of inflaming tensions further with a mock dive celebration in front of Moyes, who had criticized the striker for exaggerating challenges.
Suarez added a second quickly afterwards, nodding in a fine Steven Gerrard free kick to send the traveling fans into ecstasy after just 20 minutes with the home defense nowhere to be found.
The two-goal cushion was short lived though. Fellaini went close with a deflected shot and the following corner saw Leon Osman fire the home side back into contention. Jones could only punch the ball to the edge of the area, and the midfielder struck in from 20 yards courtesy of a flick off of Joe Allen.
Suarez almost added another with a rasping 25-yard volley, but Everton drew level after 35 minutes. After winning a contentious throw-in, Kevin Mirallas' cross hit Fellaini, but the tall Belgian reacted well to hook in a low ball which Steven Naismith tapped in unmarked from two yards out.
Mirallas' fine display continued as the winger almost put Everton ahead after 42 minutes, spinning away beautifully from Wisdom but injuring himself as the shot was deflected wide, and was replaced by Magaye Gueye at the interval.
Sterling had a golden opportunity three minutes after the break following a delightful pass from Enrique, but the youngster got the attempted chip all wrong, leaving an unmarked Suarez furious on the penalty spot.
Martin Skrtel did brilliantly to deny Nikica Jelavic on the hour mark as Coleman passed in from the right after fine link play with Fellaini, and the striker could only head wide after 65 minutes following a great Baines delivery.
Gerrard could have won it for the visitors with eight minutes remaining but his drive from the edge of the area was well blocked by Jagielka, and Suarez thought he had won the game in the 93rd minute, but his finish was chalked off on a very questionable decision.
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