Thomas 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.
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 of United 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.