Tuchel’s side dominated for long periods at Stamford Bridge but fell behind when Jorginho’s poor control presented Sancho with the chance to score early in the second half.
Jorginho made amends for that mistake as he stroked home a penalty after Aaron Wan-Bissaka fouled Thiago Silva.
It was the least Chelsea deserved after they laid seige to United’s goal and saw David de Gea make a series of fine saves.
Chelsea, unbeaten in their last 11 games in all competitions, remain top of the Premier League but are now just one point ahead of second-placed Manchester City, who defeated West Ham earlier on Sunday.
“I’m absolutely happy with the performance. We dominated from the first minute, we were aggressive and never stopped attacking. Unfortunately we scored two goals, one for each side,” Tuchel said.
“United defended deep. We have to swallow the result. We feel disappointed as we think by far we did enough. It can happen in football.”
United, in eighth place, are 12 points adrift of the leaders but this gritty display was something to build on after a traumatic period.
Michael Carrick was in charge for a second successive game after Ole Gunnar Solskjaer’s sacking, with the appointment of Ralf Rangnick as interim manager until the end of this season expected to be confirmed shortly.
United are in discussions with Rangnick, who is working as director of sports and development at Lokomotiv Moscow.
But in the meantime Carrick had overseen a vital win at Villarreal in the Champions League on Tuesday and he used the same defensive blueprint to frustrate Chelsea.
“I’m proud of the players and the group all week. It’s a difficult situation and we tried to make the best of it,” Carrick said.
“But there is that little bitter feeling. I’m a little disappointed, I can’t lie. I don’t think it was a penalty at all.”
Given Rangnick’s preference for a intense pressing game-plan that may not suit Cristiano Ronaldo, it was intriguing to see the sometimes diffident Portugal striker left on the bench by Carrick until the 63rd minute.
With Bruno Fernandes as the false nine and Marcus Rashford and Sancho on the flanks, United set up for a point and got exactly that.
Hakim Ziyech found space in the United area early on but his low shot was saved by De Gea, setting the tone for the rest of the afternoon.
Moments later, Marcos Alonso’s header from Silva’s long pass caught Victor Lindelof and Wan-Bissaka out of position.
Callum Hudson-Odoi seemed certain to score when he pounced on the loose ball, only for De Gea to repel the forward’s close-range effort with a superb save.
Antonio Rudiger was left in acres of space when his 25-yard drive was tipped onto the crossbar by De Gea.
United had ventured into the Chelsea area only twice in the first half, but they broke out to snatch the lead five minutes after the interval with their first shot on target.
Fernandes’ hoofed clearance should have been easy for Jorginho, but Chelsea’s captain made a hash of dealing with the high ball and Sancho was rewarded for pressing hard as he stole possession, sprinted clear on goal and slotted past Edouard Mendy.
After going 14 games without a goal following his close-season move from Borussia Dortmund, Sancho has two in his last two matches.
Chelsea’s response was emphatic and their pressure was finally rewarded in the 69th minute after Wan-Bissaka’s flailing challenge on Silva conceded a needless penalty.
Jorginho had missed a high-profile penalty for Italy against Switzerland recently, but the midfielder coolly sent De Gea the wrong way from the spot.
In a frenetic finale, Rudiger could have won it in the final seconds when he volleyed over from Christian Pulisic’s cross.