United went into the game on the back of the 2-2 midweek draw against Burnley which ended their eight-game winning run since Ole Gunnar Solskjaer took over from Jose Mourinho in mid-December.
However, normal service resumed as Rashford, who was celebrating his 100th appearance for the Red Devils, scored the only goal of the game inside the opening 10 minutes at the King Power Stadium.
Leicester pressed for a share of the spoils, having impressed by earning a 1-1 draw at leaders Liverpool during the week, but United held firm for the victory which takes them above Arsenal into fifth spot, just two points behind Chelsea in fourth, while the hosts remain down in 11th place.
Harvey Barnes was restored to the Leicester line-up with Marc Albrighton dropping out altogether in just one change to the Leicester side which grabbed that draw at Anfield.
Solskjaer made four changes to the side which snatched that late draw against the Clarets as Eric Bailly replaced Phil Jones, Ander Herrera came in for Andreas Pereira, Jesse Lingard replaced Juan Mata and Alexis Sanchez was given his first Premier League start under the Norwegian coach as Romelu Lukaku dropped to the bench.
United should have taken the lead in the fifth minute when Rashford got between Ben Chilwell and Harry Maguire to meet a Luke Shaw cross from the left but somehow guided his header over from six yards.
The visitors continued to hold the upper hand and Rashford tested Kasper Schmeichel with a low angled drive from a Lingard corner in the eighth minute before putting his side in front a minute later.
Paul Pogba picked up Ricardo Pereira’s clearance midway inside the Leicester half before chipping a wonderful pass into the area for the England striker who finished with aplomb past Schmeichel.
Leicester did muster a response on 15 minutes with James Maddison getting a sight of goal on the edge of the area only to be thwarted by an excellent block from Shaw.
The Foxes continued growing into the game after being dominated for the opening quarter, although they found it difficult to break down the visitors in the final third.
Former Reds’ defender Jonny Evans had a half-chance when he got up to meet a corner on 39 minutes, but Bailly did just enough to put him off as he sent his header wide of the far post.
At the other end, Rashford got another effort in from the edge of the box in stoppage time, but it was straight down the throat of Schmeichel.
Leicester started the second half on the front foot and Barnes flashed an effort across the face of goal on 48 minutes, before Maddison’s follow-up strike slammed into the side-netting.
On 59 minutes United produced a good passing move which ended with Shaw teeing up Rashford, but this time the forward flashed his strike wide of the upright.
Leicester went back to the other end and won a free-kick which Maddison fired into the wall before Jamie Vardy tried an acrobatic attempt from the rebound which David De Gea needed two attempts to save.
On 77 minutes substitute Rachid Ghezzal almost levelled with a superb free-kick which was brilliantly tipped over by De Gea at full stretch.
Substitute, Lukaku, went close to sealing the points in the 87th minute when he got free down the right of the area, but the Belgian striker’s powerful strike towards the near post was beaten away by Schmeichel.
Leicester went upfield through Barnes and he picked out Vardy from the left edge of the area, but the striker’s first time strike was again straight at De Gea.
It was end-to-end action in the closing stages with United breaking quickly and Anthony Martial was picked out down the left of the area, but his strike was too close to Schmeichel who pushed it away from his goal.
Maguire had the final chance of the game in stoppage time when a cross bounced through to him in the area, but he couldn't get his foot over the ball and it bobbled wide of the far post.