Both sets of players enjoyed periods of pressure and possession in the capital, but the defences in blue and red stood firm as Chelsea's and United's star-studded attacks struggled to impose themselves on the game.
A relatively tame first half was overshadowed by a controversial penalty incident as the ball struck Callum Hudson-Odoi's hand, but after a VAR review - in which several players were observing the screen just behind Stuart Attwell - a spot kick was not given.
Chelsea had the better opportunities throughout the game, as Olivier Giroud was only a whisker away from connecting with Hudson-Odoi's cross in the first 45, while Hakim Ziyech's first-time effort on the 49-minute mark was excellently saved by David de Gea.
Tuchel has now extended his unbeaten streak in charge at Stamford Bridge to nine matches, while Ole Gunnar Solskjaer's visitors have also strung together a nine-game run without losing in all competitions following Sunday's stalemate.
The Blues stay fifth - one point behind fourth-placed West Ham United - with that draw, while second-placed Man United are 12 adrift of runaway leaders Manchester City.
10th min: Callum Hudson-Odoi attempts to pick out Olivier Giroud at the near post, but Victor Lindelof manages to poke it behind for a corner, which comes to nothing.
14th min: Scott McTominay just gets to the ball first ahead of Mason Mount and wins Man United a free-kick some 35 yards out. Marcus Rashford gets it over the wall but Edouard Mendy calmly punches it away.
15th min: Hakim Ziyech finds himself in space and drills a low shot through a crowd of red shirts, but David de Gea safely gathers.
16th min: VAR goes back to review a potential handball by Hudson-Odoi in the area, but Stuart Attwell controversially does not point to the spot.
22nd min: Chelsea break on the right-hand side and Cesar Azpilicueta finds Hudson-Odoi just inside the area - the Englishman takes a touch before sending a powerful half-volley wide of the post.
32nd min: An awful attempt at a clearance from De Gea lands at the feet of Giroud outside the area, but the Frenchman blasts the ball over the bar.
36th min: Antonio Rudiger plays a delightful cross-field ball out to Hudson-Odoi, and Giroud is only inches away from getting a touch on the Englishman's cross.
49th min: Magnificent opportunity for Chelsea as Ben Chilwell plays it across for Ziyech to hit it first time, but De Gea sticks a hand out to save before substitute Reece James's follow up is blocked.
61st min: Sustained United pressure sees Aaron Wan-Bissaka find McTominay inside the area, but Mendy gets down well to save the Scottish midfielder's first-time strike.
63rd min: Mount gets away brilliantly but McTominay blocks his effort, and Chilwell's follow-up is stopped on its way by an offside Giroud.
68th min: Chelsea fail to close down Fred on the edge of the area and the United man gets a shot away, but the ball curls just wide of the post.
74th min: Ziyech plays a delicate pass to Mateo Kovacic, but the Croatian midfielder sees his curling effort land in the grateful arms of De Gea.
83rd min: Timo Werner almost gets onto the end of Reece James's inviting cross, but Victor Lindelof stabs a toe out to take it away from the Chelsea forward before he can get a shot away.
90th +3 min: Fred gets a last-ditch shot away to win the game for United, but Mendy parries it to safety and the final whistle blows.
MAN OF THE MATCH - ANTONIO RUDIGER
On a day where both sets of attackers flattered to deceive, Chelsea centre-back Antonio Rudiger stood firm and enjoyed an impressive display in the heart of the Blues' defence.
The German is clearly relishing having his compatriot Tuchel in the dugout and looked bright throughout the 90 minutes, making several crucial blocks and a couple of promising forays forward.
A controversial moment in the first half saw Hudson-Odoi appear to handle the ball in the area while under pressure from Greenwood, but after a VAR review, referee Stuart Attwell opted against pointing to the spot.
The decision divided opinion among pundits and fans alike amid the new handball rule, and the fact that several in blue were also observing the VAR screen, but Attwell eventually waved away the protests.
Possession: Chelsea 57%-43% Man United
Shots: Chelsea 18-11 Man United
Shots on target: Chelsea 6-4 Man United
Corners: Chelsea 3-4 Man United
Fouls: Chelsea 11-12 Man United
CHELSEA (3-4-2-1): Mendy; Azpilicueta, Christensen, Rudiger; Hudson-Odoi (James 46'), Kante, Kovacic, Chilwell; Ziyech (Werner 78'), Mount; Giroud (Pulisic 65')
MAN UNITED (4-2-3-1): De Gea; Wan-Bissaka, Lindelof, Maguire, Shaw; Fred, McTominay; James, Fernandes, Rashford; Greenwood (Martial 78')