United may have felt this match was theirs to be won had VAR and Stuart Attwell awarded a penalty early on for Callum Hudson Odoi's handball in the box.
For Ole Gunnar Solskjaer and Man United, this result now means they have now failed to beat a top-six side this season - drawing five out of seven top-six games with a result of 0-0.
Although, the Reds have scored the second-highest amount of goals this season they have remarkably failed to score in their last six games against the top six.
A positive note for United sees them now having stretched their unbeaten away record to 20 games. Solskjaer's side is now a point adrift of 3rd place Leicester City who failed to pick up points today against Arsenal.
For Chelsea, this result extends their unbeaten run to ten games since Thomas Tuchel's arrival at the club. But as a result of today's result, they now drop into fifth place and stay six points behind second-place United.
Story of the Game
The first half of the game saw chances limited for both sides and it was Tuchel’s Chelsea who was the side that dominated with their fluid passage of plays.
VAR was the talk of the town once more as Odoi's handball in the penalty area was not penalised by referee, Stuart Atwell.
United's chances in the first half were hugely limited but the breakaway chance for United came when Marcus Rashford’s over the top pass to find Mason Greenwood in the box was crucially cut out by Ben Chilwell.
Hudson Odoi caused more problems for United this time in their half, the Englishman managed to break free into United’s half but a strike from distance on the 21st minute failed to test David De Gea.
United defensive inabilities to defend crosses were very nearly shown by the Blues ten minutes before the break. Olivier Giroud and Hudson Odoi combined, but the wing-back's early cross into the Frenchman was inches away from headed contact.
United dominated the last 15 minutes of the first half after being outplayed earlier on but Solskjaer’s Reds failed to test Chelsea’s number 1, Edouard Mendy.
The second half began brightly for the hosts as they created the first big chance of the game and could’ve taken the lead three minutes in.
Chilwell's surging run down the left-hand side saw his cross into the box find Hakim Ziyech. And his strike at goal in the six-yard box was excellently saved by United’s De Gea, even though the Spaniard was seen to have been going the opposite direction of the Morrocan's shot.
Chelsea was the better side in the first twenty minutes of the second half as they had much of the ball and controlled the tempo well. But United finally came alive for the first time and created a flurry of chances all in space of a minute.
Excellent link-up between Daniel James and Greenwood on the 58th minute saw the 19-year-old's fierce strike from just outside the box go inches wide.
United created another chance as Wan Bissaka’s cut-back pass into Scott McTominay saw the Scotsman's curling strike denied by Mendy at his near post.
The last ten minutes of the game saw both sides go for the winner. Reece James's cross into Timo Werner on the 83rd minute saw Lindelof excellently defend to just gets in front of the German and crucially poke the ball out of play.
United could’ve very well scored the winner in classic counter-attack fashion on the cusp of stoppage time.
The Reds were on a three on four attack but Mctominay's low pass towards Anthony Martial and Rashford was under-hit, allowing N'golo Kante to make a crucial interception in the six-yard box.
Next time out
United's next game sees them travel to South London to face Roy Hodgson's Crystal Palace in search of a win that cements their position in the top two after now falling 12 points behind Pep Guardiola's Manchester City.
Tuchel's Blues travel to Merseyside in their next Premier League fixture to face off against Jurgen Klopp's Liverpool at Anfield - a ground Chelsea has failed to win at in the league since 2014. A game that will have huge implications in the race for the top four.