Rashford cushioned Paul Pogba's through ball into his path before blasting home his sixth goal in Ole Gunnar Solskjaer's 10 games in charge just nine minutes in.
A 2-2 draw at home to Burnley in midweek is the only match Solskjaer has failed to win and another victory lifts United a point above Arsenal and to within two points of fourth-placed Chelsea in the fight for Champions League football next season.
However, as is often the case on the road, United relied on goalkeeper David de Gea for all three points as a Leicester side that have often troubled the top six this season improved after half-time.
Twice the Spaniard stood tall to deny Jamie Vardy, but De Gea's best save came from Rachid Ghezzal's free-kick that arrowed towards the top corner.