England international Rashford scored his eighth goal in 13 outings to earn a 1-0 victory at the King Power Stadium.
Solskjaer's 10th game in charge brought a ninth win, with revived United not needing to leave third gear.
Paul Pogba and Rashford combined to seal another victory and the pair now have 12 goals under Solskjaer having struggled so badly under Jose Mourinho.
United are now just two points behind fourth-placed Chelsea. Re-live the action below:
Welcome to The Independent's live coverage of Sunday's offering of Premier League football, and what a day it could be.
We start with coverage of Leicester vs Manchester United, two teams in need of a win for different reasons. Claude Puel is continually having top prove himself to the fans and a home win against United would go a long way to gaining more support from those in the stands.
Ole Gunnar Solskjaer's side, meanwhile, are looking to provide a positive response to their 2-2 draw against Newcastle at Old Trafford in midweek. A win would keep United's top four hopes alive, something that could go a long way to deciding whether Solskjaer gets the job on a permanent basis.
Leicester City vs Manchester United
As Ole Gunnar Solskjaer looks to continue his unbeaten run as Manchester United caretaker boss he comes up against a tough opponent in Leicester City.
However Leicester have won just one of their last 21 league and cup games against Manchester United, drawing four and losing 16.
That lone victory was a 5-3 win at home in Sept. 2014, so things are looking promising for the United boss.
Leicester make one change from the team that drew at Anfield in the week as Harvey Barnes replaces Marc Albrighton.
New signing Youri Tielemans doesn't even make the squad.
As for Man Utd, Ole Gunnar Solskjaer makes four changes to the side that drew with Burnley.
Alexis Sanchez, Eric Bailly, Jesse Lingard and Ander Herrera return to the starting line up with Romelu Lukaku and Anthony Martial on the bench.
Marcus Rashford makes his 100th Premier League appearance today.
At 21 years and 95 days old, he is the youngest player to reach a century of top-flight appearances for Manchester United since Ryan Giggs in December 1993.
Anthony Martial signed a new five year deal with Manchester United on deadline day, here's what Ole Gunnar Solskjaer had to say about the 23-year-old:
"He's a fantastic finisher, great ability, he's young, he's intelligent, he knows his football.
"He knows that I'd like him to do a couple more runs in behind, but I've not seen many players with the quick feet and the skills in the last third as he has, so we're going to build his career and hopefully he'll grab the chance with both hands."#
Leicester have a good record against the top six teams this year. They're unbeaten in games against Chelsea, Manchester City and Liverpool so they can create a few problems for Manchester United today.
Will they cause another upset or will Solskjaer's Man Utd resurgence continue?
Leicester 0 - 0 Man Utd
Man Utd start the match, Ole Gunnar Solskjaer's men are unbeaten in nine matches. Can they make it ten today?
3 mins: Leicester 0 - 0 Man Utd
Sloppy work from Man Utd in the middle of the park give Leicester a corner. The ball comes into the centre of the box, skims Ndidi's head and goes out for a goal kick.
6 mins: Leicester 0 - 0 Man Utd
Man Utd win a couple of throws on the right,Young passes it out to the left with Luke Shaw. Shaw is just outside the box in space with no pressure on him, He puts the ball into the far post with his left foot and Marcus Rashford runs through tow defenders and has a free header on goal. Schmeichel doesn't get across in time but Rashford heads it over the bar!
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