The Toffees went into the game with a comfortable eight-point lead over their neighbours and should have had the presence of mind to brush aside Blackburn Rovers and consolidate their hold on fourth place.
The start of the game was marred by a series of injuries as Rovers tried to assert their presence with some tough challenges.
But it was Paul Dickov who found himself on the wrong end of a run-in with Nigel Martyn in the 15th minute and he had to leave the field for treatment with Stead coming on to replace him.
David Moyes, clearly growing frustrated by the lack of fluid build-up play, brought on Duncan Ferguson in the 24th minute for the injured Mikel Arteta.
Everton seemed to gather some energy from Big Dunc and started to threaten Blackburn's goal for the first time.
Marcus Bent found himself in a one-on-one situation with Brad Friedel, but the visitors' keeper stood tall to make the block.
The closest the Blues got to a real chance in the first half was a Lee Carsley free-kick on the edge of the box. His shot left Friedel frozen to the spot, but it was again off target.
The game continued in the same scrappy vein after the half-time break with a tussle inside Rovers' area, with several players ending up on the floor but the ball wouldn't fall for Everton.
Blackburn seemed content to disrupt the game's tempo with a steady succession of fouls and grind out a draw.
But to Everton's credit, they stayed hungry for the win and kept battling to find some rhythm.
And they came close to breaking the deadlock when Leon Osman turned Rovers' defence inside out and his soft flick floated towards the waiting head of Ferguson, but his effort was easily smothered by Friedel.
Then when the Goodison faithful thought their luck couldn't get any worse, Blackburn managed to put together their first move of the game like a bolt out of the blue and thrust a dagger into Everton's heart.
Steven Reid made a run through the midfield and found a gaping hole in the defence to thread a deft pass through to the feet of substitute Stead in the 70th minute
He only had the keeper to beat and his quick, right-footed effort left Martyn floundering on the floor.
Stead proved he was a true villain to the Goodison faithful when he was booked five minutes later for a harsh challenge on Carsley.
The home side struggled to try and salvage the game, but it was too late and they must now focus on the last ten games.