Manchester United were left frustrated after a resolute performance and the woodwork helped Everton earn a point.
Cristiano Ronaldo hit the upright with a fizzing drive, as did Alan Smith with a similar effort and Paul Scholes also saw a shot come back off the post.
But United were lucky not to concede a penalty when Mikael Silvestre blatantly used his arm to clear the ball from a lurking Duncan Ferguson.
In the first half, Louis Saha missed a great chance when he headed wide.
It was the only clear-cut opportunity of the opening 45 minutes, with United lacking the cohesion to break down an Everton side content to sit deep and defend.
Smith and Saha were isolated up front for the home side, while Marcus Bent was left to lead the line on his own for Everton.
How the game could have done with Wayne Rooney, the striker set to leave Everton and possibly join United.
But in truth, the problem for United was their lethargic midfield with Kleberson largely ineffective and Scholes lacking his usual influence.
But Everton's Tim Cahill was busy and the Australian brought a smart save from Tim Howard, after latching on to Hibbert's through ball.
With the ball more often in the air than on the grass, chances were few.
Everton's defence was standing firm, and was impressively marshalled by a bandaged David Weir, who suffered a nasty cut after a clash with Smith.
The visitors' Leon Osman fired over the bar after Bent had robbed a hesitant Silvestre, while United's Gary Neville blazed over after cutting inside Alessandro Pistone.
Saha should have opened the scoring when Kleberson's cross found him unmarked in the area, but the Frenchman directed his header wide.
After the break, United found another gear and started to apply the pressure as they searched for a breakthrough.
But each time United had beaten Everton keeper Nigel Martyn, the goalpost denied them.
Ronaldo, Smith and Scholes all saw shots come back off the post, while Smith also went close with a glancing header late on.
Everton twice made appeals for a penalty, the first when it looked as though John O'Shea's arm prevented Cahill reaching Bent's pass.
The second appeared more clear, and Silvestre was lucky to have got away with handling the ball off the top of Ferguson's head.
Everton almost gifted United a goal when Alan Stubbs' header had to be tipped over the bar by a scrambling Martyn.
But the Toffees held on to snatch a vital point.
Man Utd: Howard, Gary Neville, Silvestre, O'Shea, Spector,
Fletcher (Giggs 64), Scholes, Kleberson (Djemba-Djemba 64),
Ronaldo (Bellion 81), Smith, Saha. Subs Not Used: Phil Neville, Carroll.
Everton: Martyn, Hibbert, Weir, Stubbs, Pistone, Osman, Watson,
Carsley, Kilbane, Cahill (Naysmith 70), Bent (Ferguson 54).
Subs Not Used: Wright, Campbell, McFadden.
Booked: Osman, Cahill.
Referee: D Gallagher (Oxfordshire).