Gabriel Agbonlahor's late equaliser secured a draw for Aston Villa in a game that Everton dominated.
Leon Osman put Everton ahead with a 12-yard finish and they had a penalty shout turned down when Barry Bannan appeared to foul Leighton Baines.
Villa levelled through a stunning, 30-yard strike by Stiliyan Petrov, but Baines's penalty made it 2-1 after Fabian Delph fouled Phil Jagielka.
But Agbonlahor's header from Marc Albrighton's cross earned Villa a draw.
Everton, playing their first game since Mikel Arteta's departure to Arsenal, had gone into the game without a recognised striker. They created chances aplenty though, with Tim Cahill leading the line in front of Leon Osman, and should have scored more.
Moyes encouraged by team performance
The performance would have heartened the home crowd with some of their fans having protested at chairman Bill Kenwright and the board before kick-off. There was some warm applause for Kenwright, however, when he was shown on the big screen during the game.
Villa would have been grateful to their defence, particularly the imperious Richard Dunne, for keeping Everton's goal tally down. Referee Michael Oliver also helped them by waving away the home penalty appeal at the start of the second half when Everton were 1-0 ahead.
Everton had asserted themselves quickly as an early Jagielka header was cleared off the line by Darren Bent and Osman was then denied by Villa goalkeeper Shay Given from close range.
Jagielka was frustrated again with another header as his effort struck the side-netting following a Baines corner.
But Everton finally got what they deserved in the 19th minute after a patient passing move that culminated in Tim Cahill dragging the ball back for Osman to tuck the ball in.
McLeish reflects on a fortunate point
Everton finished the first half strongly as Marouane Fellaini and Jack Rodwell both came close with 25-yard strikes. Villa only had a 35-yard shot from Bannan to show for their lacklustre efforts.
Bent squandered a chance when he mis-hit from eight yards just after half-time, but Everton continued to look the more dangerous team.
David Moyes's men were disappointed when Bannan appeared to trip Baines, after an impressive run by the left-back, but no penalty was given.
Villa were level against the run of play just after the hour when Petrov unleashed his long-range effort with Everton's defenders failing to identify the danger.
But the home side looked to have made amends, six minutes later, as they were awarded a penalty during a scramble when Delph climbed on Jagielka, with Baines scoring from the spot.
The Midlands team levelled again unexpectedly when Albrighton's cross from the left was met by Agbonlahor at the back post with seven minutes to play.
Everton had further chances to win the game late on, but Villa survived.
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