The Portuguese made it 10 matches against Wenger as Chelsea manager without suffering defeat as Arsenal saw their winless Premier League run extended to three games.
Wenger's side were left aggrieved by some contentious refereeing decisions in each half, although it was Mourinho's men who arguably had the better chances to take all three points.
Still, Arsenal would have claimed the three points had Olivier Giroud not wasted an excellent chance to seal victory in the latter stages of the second half.
And Wenger was left to rue what might have been as the result left Liverpool top on goal difference, with Chelsea moving into fourth.
The Frenchman made four changes from the side beaten 6-3 by Manchester City last time out, Thomas Vermaelen replacing the injured Laurent Koscielny in defence, while Tomas Rosicky and Mikel Arteta came in for the suspended Jack Wilshere and Mathieu Flamini in midfield.
Frank Lampard was restored to the Chelsea line-up as part of three alterations made by Mourinho, who also selected Gary Cahill and John Obi Mikel in his starting XI.
The game got off to a slow start and it was the visitors crafted the first opening as Ramires headed over from Willian's cross after 25 minutes.
Chelsea began to exert more influence as the half progressed, and Mourinho's charges were unlucky not to take the lead just after the half-hour mark when Lampard's half-volley crashed against the underside of the crossbar.
Fortune looked to favour Chelsea in the latter stages of the first half, first Mikel appeared to catch Arteta hight with a sliding tackle, only to escape punishment from referee Mike Dean, who then waved away claims for an Arsenal penalty following Willian's apparent foul on Theo Walcott.
Chelsea were again perhaps fortunate to keep 10 men on the field 18 minutes into the second half as Ramires was booked for a rash challenge on Arteta.
Tensions threatened to boil over in what was an ill-tempered affair, Mesut Ozil and Branislav Ivanovic coming together soon after the Chelsea defender had forced a save from Wojciech Szczesny with an effort from range.
Arsenal almost claimed the victory 11 minutes from time, but Giroud spurned a good opening as he fired into the side-netting from the near post.
And Bacary Sagna saw a late header cleared off the line as Arsenal finished strongly without managing to find a way past Petr Cech.
