The Uruguayan put the visitors in the driving seat but Brendan Rodgers' side threw away a two-goal cushion as David Moyes' men responded well to claim a point at Goodison Park
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.
The Reds were left bemoaning a controversial decision by the officials in injury time, however, when Suarez appeared to be incorrectly flagged for offside after beating Tim Howard from close range.
Marouane Fellaini returned for the hosts after recovering from a knee injury, while Kevin Mirallas took the place of the suspended Steven Pienaar on the left wing.
Brad Jones continued in goal for the visitors despite Pepe Reina being fit enough to take a place on the bench, and Jose Enrique started as the injured Glen Johnson missed out.
After a slow start the game sprang into life after 13 minutes when the visitors took 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 criticised the striker for simulation before the match.
Suarez added a second quickly afterwards, nodding in a fine Steven Gerrard free kick to send the travelling fans into ecstasy after just 20 minutes with the home defence nowhere to be found.
The two-goal cushion was short-lived though - Fellaini went close with a deflected shot and the subsequent corner saw Osman fire the hosts back into contention. Jones could only punch the ball to the edge of the area, and the midfielder struck from 20 yards courtesy of a deflection off 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 Naismith tapped in unmarked from two yards out.
Mirallas' fine display continued as the winger almost put the hosts ahead after 42 minutes, spinning away beautifully from Wisdom but injuring himself as the shot was deflected wide. He 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 his attempted chip all wrong, leaving an unmarked Suarez waiting in fury on the penalty spot.
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. Suarez thought he had gone one better in the 93rd minute, but stabbed finish from another Gerrard free-kick was ruled out by the assistant referee.
Sebastian Coates climbed on the back of Jagielka to win the initial header but it was unclear whether the official had stopped play for a foul or for offside, with Suarez in line with the last defender.