Wenger still winless against Jose
The Gunners boss was bidding for a first victory over his Portuguese adversary at the 10th attempt, knowing that was what his side required to return to the top of the Barclays Premier League at Christmas.
But a defensive-minded Chelsea claimed a draw which took them fourth in the standings and Arsenal moved second, behind leaders Liverpool on goal difference.
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.
After 30 minutes of sizing each other up, the contest ignited when Frank Lampard hit the bar and the tackles began to fly in.
Willian passed up a good opportunity on the counter-attack for a Chelsea side which had set up defensively, putting men behind the ball in an effort to limit the space Arsenal's creative players, such as Mesut Ozil, could work in.
Arteta's right ankle took another hefty blow from Ramires, who was booked, as the second half continued in much the same vein as the first.
Olivier Giroud sliced wide with 11 minutes remaining and was denied by Petr Cech at point-blank range with five minutes to go as the honours ended even, with Mourinho the happier of the managers.
Both sides entered the contest on the back of a defeat.
The Blues' former Arsenal left-back Ashley Cole, who attended his former club's Christmas party last week, was named among the substitutes.
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, Rosicky and Aaron Ramsey.
Despite their defensive approach, it was Chelsea who threatened first when Torres' attempt to burst through was brought to an end by a cynical challenge by Per Mertesacker, who was fortunate not only to avoid conceding a free-kick, but also a yellow card.
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.
Mourinho questioned fourth official Jon Moss to see if the ball had crossed the line, but the officials' watches did not buzz and replays showed the ball bounced well clear.
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 Dean deigned no foul had been made.
Willian should have done better at the other end than shoot weakly at Szczesny after Ramsey conceded possession to allow Hazard to counter.
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 of the near post from a position to the left of goal, 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 and made it clear a point was more than acceptable.
Arsenal had a final chance to claim three points, but Azpilicueta cleared a Vermaelen header from a corner off the line.
Arsenal fans finished by labelling Chelsea "boring" as they were left frustrated.
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