Both sides pressed forward looking to arrest the stalemate, but neither was able to provide a clinical finish, as numerous chances were squandered, with Everton striker Jo the main culprit.
The Brazilian striker had three glorious opportunities to break the deadlock, while Didier Drogba hit the crossbar for the home side in a frantic finale.
The visitors contributed significantly to the match as a whole, defending stoutly throughout to frustrate the hosts.
The Toffees appealed vehemently for a first-half penalty when Leighton Baines appeared to be tripped by Alex, but referee Mark Halsey was unconvinced.
Indeed, Baines was rightly awarded man-of-the-match after an inspired rear-guard by the visitors was led by the left-back and Joleon Lescott, in particular.
Blues boss Guus Hiddink opted to name an unchanged side from the FA Cup semi-final victory over Arsenal, while David Moyes rested Marouane Fellaini and Phil Jagielka as he freshened up his starting line-up.
While the home side pressed as though the onus was firmly on them to create an opening, the gaps which emerged in the first half were at the other end.
Everton broke on the counter attack with real conviction, but without the finish to match, while Chelsea dominated possession for extended periods.
Steven Pienaar whipped in a devilish cross from the left inside the opening five minutes, which found its way through to Jo in the centre, but the striker failed to get the ball under control and it squirmed out to Petr Cech, who was relieved to collect it.
Two minutes later, Didier Drogba almost found his way through on goal after Joseph Yobo defended like he was standing in quick sand, but his centre-back partner Lescott spared his blushes with a covering block.
Still inside the opening 10 minutes, Jo squandered a glorious chance with just Cech to beat, but the keeper spread himself to save with his legs and deny Everton from taking an early lead.
The Brazilian really ought to have scored after he latched on to Pienaar’s pass, with Ashley Cole playing him onside from left-back.
Jo then dallied with the ball out wide on the left wing as Everton broke on the counter-attack, but Tim Cahill still managed to turn Alex with the pass played behind him, but his shot was parried away by Cech on the half-hour.
Everton had their vociferous appeals for a penalty rejected by referee Mark Halsey, as Baines scampered into the Chelsea box before the left-back appeared to have been tripped by Alex, who hung out a leg. But crucially, referee Halsey was unconvinced.
With half-time looming, a pacey break from Anelka directly following an Everton corner almost caught the visitors out, but the Frenchman’s pass to his compatriot Malouda was pounced upon eagerly by Howard.
Jo then lost his footing at the other end, as another clear-cut opportunity fell to the striker, which left him shaking his head in dismay on the stroke of half-time.
The second half took a while to settle down, with both sides frantically chasing the ball and subsequently lacking in composure when in possession, and Moyes was clearly the happier manager on the touchline.
Malouda enjoyed a foraging run down the left in the 55th minute, which culminated in a cut-back for Drogba, who was poised to unleash a shot from close range, but for a timely challenge by Lescott.
Five minutes later, and a rasping drive from John Terry on his weaker left foot brought a sharp stop from Howard, diving to his left to palm the ball away to safety.
Chelsea were then left incensed by a clumsy challenge by Yobo on Malouda, who went down clutching his face after both players challenged for a high ball, but the Nigerian was not penalised.
In the 73rd minute, Jo hooked another wasteful shot over the bar after he and Pienaar had created space on the edge of the Chelsea area – but is was not to be the Brazilian’s night.
After an innocuous challenge, Cech and Jo collided, with both going down clutching their heads, and the
Chelsea and Everton shared a 0-0 draw in the Premier League after a frustrating evening of dogged defending and missed chances at Stamford Bridge.
Brazilian’s frustrating evening was finally brought to an end, as he was one of a flurry of late substitutions.
Salomon Kalou then came close for Chelsea with a header from close range which went over the bar after Lampard had drilled a shot which was blocked by a combination of Lescott and Yobo.
As the game drifted to its conclusion, Drogba had the final chance; the Chelsea striker struck the bar with a sharp swivel and shot from close range, but it summed up a match which was dominated by two obdurate defences, and less than clinical finishing.
The draw moves Moyes’s side to within one point of fifth-placed Aston Villa, while Hiddink will be able to focus on his team’s cup exploits.
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