Chelsea earned their fourth clean sheet in five games however, there is still a clear sense of disappointment as a victory would have propelled Thomas Tuchel's men into the top four.
The Blues came close to breaking the deadlock in the second-half when Hakim Ziyech's effort was saved by David de Gea.
United too had their opportunities. Scott McTominay, Fred and Bruno Fernandes all threatened Edouard Mendy only to come out empty-handed.
As often the case in goalless draws, the two goalkeepers performed best in fantasy football. Both De Gea and Mendy received ten points after a strong outing in London.
Andreas Chistensen was the third highest scorer with eight points to his name. The 24-year-old defender was given two bonus points after a largely faultless performance.
From the managers
Chelsea boss Thomas Tuchel told Sky Sports: "We had the better chances and played high up the pitch in the second half. We have to accept a draw in the end and a clean sheet. We had a very good reaction after half time. We stepped up and got better and better with higher recoveries. We went for a win but it did not happen."
Manchester United manager Ole Gunnar Solskjaer told the BBC: "It was a penalty, 100% nailed on. If that's a natural position for a hand to be in when the ball comes in then I must be blind.
"We wanted to win. Apart from the last quality or pass being not good enough... we had great pressing, desire, work-rate... There was a great save by David de Gea and we got the clean sheet.
You can find more reaction to this game on TEAMtalk: Solskjaer loses cool with Man Utd penalty denial; Shaw reveals details of bizarre referee conversation...
Chelsea's battle for the top four continues on Thursday when they clash with Liverpool. Just a few days later they take on Everton at home in what is a huge week for the Blues against Merseyside clubs.
Man United, on the other hand, return to the capital on Wednesday for a game with Crystal Palace. A few days later, the Red Devils take on the league leaders in a much-anticipated Manchester Derby.
The bookmakers slightly preferred the home side ahead of kick-off having priced Chelsea at 13/10 and United at 21/10. Nevertheless, the draw was one of the more popular bets in this game and the odds were relatively attractive at 12/5.
Looking at some of the outright betting, Solskjaer's men have taken a dive in the title market. City are currently priced at 1/200 to win back the Premier League crown while the Red Devils are 125/1.
Interestingly, Chelsea are even money to finish in the top four and looking at how things are panning out, that is a steal.