In a first half low on chances, the biggest talking point came when Callum Hudson-Odoi appeared to handle in the Chelsea penalty area as he beat Mason Greenwood to a loose ball.
A Video Assistant Referee (VAR) review saw Stuart Attwell consult the pitchside monitor, but the match official stuck with his original decision of no penalty - to the fury of United captain Harry Maguire.
Olivier Giroud had two fine openings for Chelsea but could not connect with either chance for a header, while Hudson-Odoi flashed wide on the volley.
Chelsea had the best chance of the game up until then, with David de Gea making a strong point-blank stop from Hakim Ziyech.
The game continued to open up as Scott McTominay then forced a save from Edouard Mendy not long after, while Fred and Greenwood flashed shots wide from outside the box.
However, neither side could find the quality in the final third to secure all three points.
The result will suit Ole Gunnar Solskjaer, who United going clear in second place and still managing to capitalise on defeats for Leicester and West Ham this weekend.
The point was also a boost for Chelsea given those results, but Thomas Tuchel may feel it was an opportunity missed.