Although United had the better of David Moyes' first home Premier League game in charge, they lacked the authority to exert the kind of sustained pressure which might have shaken John Terry out of his defensive superiority.
Last season's Golden Boot winner Robin van Persie barely had a chance of note, his best opportunity coming three minutes from time when his goalbound volley deflected to safety off Gary Cahill and John Obi Mikel.
Jose Mourinho has made no secret of his desire to sign Wayne Rooney at some point in the next seven days.
Playing in a deeper role than Mourinho would use him for Chelsea, the England forward did well on his first start since April, and tested Petr Cech with one late strike.
If anyone looked likely to break the deadlock it was Rooney, who was cheered by both sets of supporters.
It did not happen though as both sides were forced to settle for a point which, under the circumstances, neither will be too unhappy with.
When United last played a Premier League game at Old Trafford, Sir Alex Ferguson told the club's supporters they had a duty to get behind his replacement.
They heeded that clarion call, chanting Moyes' name at regular intervals and baiting Mourinho for "wanting his job".
It was part of an amusing exchange of crowd banter, that included Wayne Rooney being cheered by both sets of supporters, Chelsea's adding a cheeky: "We'll see you next week" after a run at Gary Cahill.