Another substitute, Marouane Fellaini, added a second late on to seal a third Premier League win in a row.
After a frustrating first half for United, who had a Juan Mata strike ruled out for offside, they were denied again shortly after the break.
Rashford finished from Henrikh Mkhitaryan's corner three minutes after coming on, just as it looked like Jose Mourinho's men - who had seen Kasper Schmeichel save Romelu Lukaku's penalty - would have to settle for a frustrating home draw that typified last season's top-flight struggles.
Jose signed for United manager and won three silverwares in his first season at Old Trafford.
The win was secured eight minutes from time when another substitute, Jesse Lingard, played the ball into the area and Fellaini flicked it past Schmeichel.
He wasn't as prolific last season with six goals and three assists but he wasn't the only one as Leicester suffered a huge title hangover.
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The Red Devils needed a little longer than the first two matches, but they eventually broke down another team which came into Old Trafford to sit back and defend, knocking off Leicester City 2-0.
You can not win every game 4-0, of course, but in a game that might have ended 0-0 last season, Manchester United still found a way to come out on top.
Mourinho made his first move by bringing on Rashford for Mata and the England worldwide had the desired impact, netting in the 70th minute.
And Mourinho said in his postgame news conference: "This is about the quality of the players because they are good".
It may be that Martial and Rashford are locked in a battle to start all season.
Mahrez, meanwhile, handed in a transfer request back in May but Leicester have so far fended off any interest form potential suitors.
United's hierarchy, though, are reluctant to part with the £50m required to persuade Leicester to sell their star forward and could force Mourinho to go with the players he has at his disposal until January at the earliest.