In a tight opening period in this London derby, Frank Lampard volleyed against the bar.
John Obi Mikel was perhaps fortunate to avoid a red card for a studs-up tackle on Mikel Arteta and, in the same move, referee Mike Dean also opted not to award a penalty when Willian tripped Theo Walcott.
After the break, Olivier Giroud shot wide and Thomas Vermaelen had an effort cleared off the line by Cesar Azpilicueta as Arsenal could not find the win they needed to retake the Premier League lead.
The Gunners are now level on points with Liverpool at the top, while the stalemate sees Chelsea move back into fourth, two points off the top.
The draw means Arsene Wenger remains without a victory over Jose Mourinho's Chelsea.
The Gunners boss was bidding for a first win over his Portuguese adversary at the 10th attempt, knowing that was what his side required to return to the top at Christmas.
Wenger has had plenty of shots at masterminding victory over Blues sides managed by Mourinho, who now holds a record of five wins and five draws against the Frenchman.
There were numerous classics in Mourinho's first Stamford Bridge spell, but the latest bout was not in their class.
Chelsea beat Arsenal on enemy territory in the Capital One Cup in October but their approach in the Premier League was less adventurous.
Mourinho made six changes following the cup loss at Sunderland, with Petr Cech, Branislav Ivanovic, John Terry, Ramires, Eden Hazard and Fernando Torres recalled.
With Ramires, Mikel and Lampard deployed in midfield, Mourinho adapted his favoured 4-2-3-1 formation in an effort to nullify Arsenal's Mesut Ozil, Tomas Rosicky and Aaron Ramsey.
Wenger made four changes following the 6-3 loss at Manchester City, drafting in Kieran Gibbs, Rosicky, Vermaelen and Arteta.
In a tight contest, few chances were created and Gary Cahill cut out a Bacary Sagna cross with Giroud lurking, while Ramires headed over after Willian delayed making a cross on the counter attack.
A delightful return pass from Hazard to Lampard nearly broke the deadlock, but the midfielder could not keep his shot low and hit the bar.
Wojciech Szczesny made his first save to stop a Torres shot on the turn before an Arsenal response in a move which might have seen Chelsea reduced to 10 men.
Arteta and Mikel both lunged for a ball in midfield and the Nigerian's studs caught the Spaniard on the ankle. Play continued and Willian fell for Walcott's trickery, hooking the winger's ankle, but referee Dean deigned no foul had been made.
Arteta was again fouled in midfield combat, this time by Ramires, who was booked, before Ivanovic enraged Ozil with a high-foot challenge, resulting in a coming together.
There was another in a competitive contest when Rosicky slid in on the Serbian right-back, who went down easily, with the Czech booked for persistent fouling.
Hazard was replaced by Andre Schurrle and Oscar came on for Willian as Mourinho recognised the need for fresh legs on the flanks to undertake defensive duties.
Cahill snuffed out an opening for Giroud as Arsenal continued to find chances limited.
Ramsey's pass then presented Giroud with a fine opportunity, but the France striker sliced wide, with the angle becoming more difficult.
Gibbs then combined with Rosicky on the left and centred towards Giroud, whose close-range effort hit the spread-eagled Cech.
David Luiz came on for Torres with three minutes to go as Mourinho shut up shop.
Arsenal had a final chance to claim three points, but Azpilicueta cleared a Vermaelen header from a corner off the line as the honours ended even, with Mourinho the happier of the managers.