In a game of few notable chances, both sides largely cancelled each other out, as Thomas Tuchel continued his unbeaten reign as Chelsea manager.
Chelsea are now unbeaten in ten games in all competitions, but a first league win over Manchester United since 2017 was not to be.
The visitors, meanwhile, had not lost in eight of their last games in all competitions, winning four and drawing as many. After the draw at Stamford Bridge, Ole Gunnar Solskjaer's men are the only team unbeaten in 13 away games in the league this campaign.
The first half began with little or no intensity from either side, except for a few half-hearted appeals for a penalty from the visitors.
Chelsea began the second half with a higher tempo but failed to pierce the well-marshalled Manchester United defence. Although the second half saw more intent from both sides, it eventually produced no goals, and both teams had to settle for a draw.
Chelsea remain fifth, one point away from the top four, while Manchester United stay second for another week, a point ahead of third-placed Leicester City.
On that note, let's have a look at the player ratings of both teams.
Chelsea player ratings
Edouard Mendy - 6.5/10
The Chelsea shot-stopper Eduard Mendy was untested for most of the first half but was called on to do more in the second half. He reacted well to deny McTominay in the second half.
Callum Hudson-Odoi - 6/10
The Chelsea youngster created the first real chance of the game with a cross from the right-wing for Giroud, which the striker could not convert. However, Callum Hudon-Odoi had to come off at half-time after sustaining an injury.
Cesar Azpilicueta - 6/10
The Chelsea captain Cesar Azpilicueta had the most touches for his side. While he was largely untroubled for most of the game, he did his bit whenever he was needed to.
Andreas Christensen - 7/10
Andreas Christensen had a Man-of-the-Match outing for Chelsea. Filling in for the injured Thiago Silva, the defender put in a calm and assured display at the heart of the Chelsea defence, completing all of his passes, duels and take- ons.
Antonio Rudiger - 6/10
Antonio Rudiger's resurgence under Tuchel continued. The German was solid at the heart of Chelsea’s defence, as he strived to drive past Manchester United attackers and switched play superbly.
Ben Chilwell - 6/10
The left-back returned to the playing XI after sitting out four straight matches on the bench. Ben Chilwell did well driving forward down his flank, creating chances for Chelsea, but he struggled to deal with Manchester United's press.
Ngolo Kante - 6.5/10
It was an utterly dominant performance from the Chelsea midfielder. N'Golo Kante was unshakeable in the middle of the park, where he broke down countless Manchester United attacks. He completed a game-high seven tackles and also registered four interceptions.
Mateo Kovacic - 6.5/10
Mateo Kovacic was his usual silky self, weaving past the Manchester United midfield with ease. Manchester United's midfielders struggled to keep up with the Croatian, who completed a team-high four take-ons.
Mason Mount - 6/10
The Englishman provided the energy in Chelsea's attack as he carried the ball forward into acres of space left behind by Manchester United's midfielders. Mount also won the most fouls for Chelsea on the night.
Hakim Ziyech - 6/10
The Moroccan struggled to get a grip of the game in the first half, where he was also guilty of some overzealous passes. However, Hakim Ziyech got better in the second half and was unfortunate not to have scored the opening goal of the game.
Olivier Giroud - 6/10
The Frenchman came close to getting the first goal of the game but failed to direct his header goalwards. Olivier Giroud's impact reduced as the game wore on before he was replaced by Christian Pulisic in the 65th minute.
Player ratings of Chelsea Substitutes
Reece James - 6/10
Reece James came on in the second half for the injured Hudson-Odoi and was very lively down the right-wing for Chelsea, putting in some useful crosses.
Christian Pulisic - 5/10
Christian Pulisic came on for Olivier Giroud in the second half. While he showed some deft touches, he eventually had no meaningful impact on the game.
Timo Werner - 5/10
Timo Werner came on late but couldn't do much against Manchester United.