"On the pitch, in the goal, I was feeling danger for nearly the whole game. We defended well, they missed big chances and the one they give us a chance we score. It is not enough to draw but at the moment it is a big point," United goalkeeper David de Gea said after the match.
"The last games before we were conceding three or four goals. We are improving. This is just two games. At least now we are defending well. We are fighting for every ball. We are a team."
Sancho was able to give United the lead on 50 minutes when a Reece James free-kick was half cleared. Jorginho miscontrolled the ball which allowed Sancho, with Marcus Rashford alongside him, to run at the goal and slot the ball home.
The result lessens Chelsea's lead at the top of the table to one point while United climb to eighth.
"We cannot be surprised [that the title race is so close," Chelsea boss Thomas Tuchel said after the game.
"Things like this happen. We played such good games against Burnley and Manchester Untied and feel disappointed as we think by far we did enough. Over a long season you drop points and you dig in and try to steal them back if someone steals them from you.
"We are in the middle of the race and this is where we want to be."