The England striker put the Red Devils in front from the spot after Daniel James was felled by Marcos Alonso inside the Blues penalty area in the 26th minute.
Chelsea hauled themselves level through Michy Batshuayi after the Belgian held off four visiting defenders to drill into the bottom corner from 20 yards.
But Rashford’s stunning free-kick ended Chelsea’s seven-match winning streak as they failed to get a measure of revenge on United for their 4-0 defeat on the opening day of the season.
The teams have gone in opposite directions since then, with United struggling for form while Chelsea are fourth in the Premier League and joint top of their Champions League group.