The England international converted from the penalty spot in the eighth minute to hand the Red Devils a huge three points, with the result moving the team into fourth position in the Premier League table.
The Foxes had their moments during the 90 minutes of action, but Brendan Rodgers's side were unusually lacklustre in the final third as the home side kept a welcome clean sheet.
United were awarded a penalty in the seventh minute, however, when Caglar Soyuncu clumsily brought Rashford to the deck, and the England international climbed to his feet before sending the ball into the bottom corner in emphatic fashion.
It was Rashford's third Premier League goal of the campaign - his second from the penalty spot, with Paul Pogba's absence through injury ensuring that there was to be no debate over the taker.
There was not too much in the way of chances in the period that followed; indeed, the next serious opportunity did not arrive until the 30th minute, when Ben Chilwell's powerful drive was tipped over by De Gea, who is said to be closing in on a new deal with the 20-time English champions.
Both teams were having their moments in the final third, although it would be fair to say that the overall quality inside the box was disappointing, with Leicester struggling to carve out clear chances at Old Trafford.
In the end, two smart saves from De Gea and a penalty from Rashford was the story of the first half in Manchester as Ole Gunnar Solskjaer's side entered the break with a 1-0 advantage.
Maddison threatened to meet a long pass from Ricardo Pereira in the early stages of the second half, but De Gea was on hand with more confident work to prevent the Englishman from having a shooting chance.
The Foxes continued to cause issues, though, and Maddison sent a dangerous free kick just wide of De Gea's goal in the 55th minute during an open period of the match in Manchester.
Harry Maguire threatened to score against his former club when he headed a corner towards goal just before the hour, although it was a comfortable save for the Leicester goalkeeper.
Youri Tielemans then struck over the United crossbar in the 61st minute during a period where the home side needed Maguire to be at his towering best with the pressure firmly on.
Maddison, having been moved into a central position, started to run the game for Leicester with the visitors firmly on top as the contest headed for its final 20 minutes.
Rashford almost put the game beyond Leicester in the 83rd minute with a brilliant free kick from distance - the woodwork came to the away side's rescue, though, as the score stayed 1-0.
It mattered little in terms of the final result, however, as Solskjaer's side held on - surviving a late chance for Wilfred Ndidi - to pick up a huge three points after a poor start to the 2019-20 campaign.
Next up for United is their first Europa League group match against Astana on Thursday night, before travelling to West Ham United in the Premier League next Sunday. Leicester, meanwhile, will attempt to return to winning ways when they host Tottenham Hotspur on September 21.
MANCHESTER UNITED (4-2-3-1): De Gea; Wan-Bissaka, Lindelof, Maguire, Young; McTominay, Matic (Fred 68'); Andreas, Mata (Chong 70'), James (Tuanzebe 91'); Rashford
LEICESTER CITY (4-3-3): Schmeichel; Ricardo, Evans, Soyuncu, Chilwell; Choudhury (Perez 57'), Tielemans, Ndidi; Gray (Barnes 68'), Vardy, Maddison