The England striker finished clinically from Paul Pogba's pass to earn United a seventh win in eight league games under caretaker manager Ole Gunnar Solskjaer.
Their winning run had been halted by Burnley on Tuesday, when the Red Devils fought back from 2-0 down to claim a dramatic 2-2 draw, but the three points rarely looked in much doubt after Rashford's 10th top-flight goal of the season.
Rashford somehow headed over from barely two yards out within the first five minutes, but he soon made amends, controlling Pogba's lofted pass expertly and drilling beyond Kasper Schmeichel.
United eased off after taking the lead and invited some Leicester pressure, although a wayward header from Jonny Evans and a dipping Pereira strike were the best openings they could muster before the break.
David de Gea was given a tougher test on the hour mark, reacting well to stop Jamie Vardy's half-volley from crossing the line after he spilled the initial attempt, and Solskjaer promptly introduced Anthony Martial for the ineffective Alexis Sanchez in a bid to spark some life into his side.
De Gea was called on again to keep out Rachid Ghezzal's dipping free-kick, while Evans swiped at thin air after being teed up by Harry Maguire, but United held firm for a 17th victory in their last 21 meetings with the Foxes in all competitions.
What does it mean? United back to within two points of Chelsea
Solskjaer's side were compelled to respond when Chelsea dispatched Huddersfield Town 5-0 on Saturday, and beating Leicester means they are back to within two points of Maurizio Sarri's side, who sit fourth.
Leicester, who held leaders Liverpool to a 1-1 draw at Anfield last time out, are 11th in the standings, six points adrift of the top seven.
Solid Shaw on song
Not many United players particularly stood out, but Shaw was a significant obstacle to Leicester down their right wing. He defended stoutly, used the ball efficiently and made one excellent block against James Maddison.
Sanchez struggles again
Sanchez worked hard down the left and made quite a few decent runs, but these went mostly unseen by his team-mates. He cut a frustrated figure when Solskjaer took him off for Martial 67 minutes in.
Key Opta facts
4 - Manchester United have won four more points than any other Premier League team since Ole Gunnar Solskjaer’s first match in charge (22 points - W7 D1 L0). Resurgent. #LEIMUN pic.twitter.com/RFYpum5WkS— OptaJoe (@OptaJoe) February 3, 2019
- Manchester United have won five consecutive away games in all competitions for the first time since October 2012.
- Leicester City have lost more Premier League games against Man Utd (17) than they have against any other side.
- Since Ole Gunnar Solskjaer’s first game in charge of Man Utd on December 22, Paul Pogba has had a hand in more Premier League goals than any other player (11 - 6 goals, 5 assists).
- Leicester have lost four of their last five games in all competitions (D1), as many as they’d lost in their previous 15 matches combined.
- Marcus Rashford has scored more goals in all competitions since his debut (42) than any other Manchester United player.
- Only Ryan Giggs (21y 74d) has reached 100 Premier League appearances for Manchester United at a younger age than Marcus Rashford (21y 95d).
- Referee Mike Dean handed out eight yellow cards in this match (four to Leicester, four to Man Utd) – only one Premier League game (Arsenal vs Huddersfield in December) has seen more yellow cards awarded this season (9 yellow cards).