The Chelsea goalkeeper, who had thwarted the United striker when one-on-one with an outstretched leg in the first half, went full length to parry away Rashford's curling strike.
Mendy had also turned behind Juan Mata's curled effort in a first half of few chances.
United substitute Edinson Cavani almost scored with his first Premier League touch just before the hour, but his flick found the side-netting.
United are still searching for their first home win of the season and stay 15th with seven points.
A fourth match unbeaten leaves Chelsea sixth on nine points.
King of the Match: Edouard Mendy (CHE)
Man Utd: 1 Nov v Arsenal (H)
Chelsea: 31 Oct v Burnley (A)