After a stuttering start to the season, Erik ten Hag's side came into the match off the back of Premier League wins over Liverpool and Southampton and saw both captain Harry Maguire and Cristiano Ronaldo left out of the starting XI for the third game in a row.
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United hit Leicester on a quick counter-attack midway through the first half as Marcus Rashford threaded the ball through the hosts' backline to Sancho, who rounded goalkeeper Danny Ward with his first touch and rolled the ball into an empty net with his second to finish the move.
James Maddison went close for Leicester with a superb free kick which brought an excellent save from United's David De Gea, but the home side struggled to create chances.
The opener seemed to boost the visitors' confidence and they pushed forward throughout the remainder of the opening 45 minutes. No second goal came and United took their 1-0 lead into the break.
Ronaldo, who did not leave the club before Thursday's end to the transfer window, came on shortly after in place of Sancho and looked lively, setting up Rashford for a chance in the 82nd minute and bouncing an overhead kick past Ward's far post minutes later.
"We need a good squad and we needed numbers," Ten Hag said after the match. "There are a lot of games to cover. Once we have Cristiano Ronaldo and Casemiro getting fitter it will get better still -- we need not only a team we need a squad."
Leicester had a late chance to level but full-back James Justin blasted high and wide from a promising position on the right as United collected back-to-back away victories.
Man United saw out the remainder of the match to move into fifth place in the Premier League on nine points -- one ahead of rivals Liverpool in the table -- while winless Leicester sit dead last with just a single point from their opening five games.
Leicester lost their key defender Wesley Fofana to Chelsea, but manager Brendan Rodgers was pleased to have replaced him with Belgian Wout Faes from Reims and to have kept Harvey Barnes and Maddison, despite reports of interest in them.
"There were enquiries for Barnes and Maddison early in the window. We couldn't lose too many. We were low in terms of depth and quality. Thankfully, they are still here," Rodgers said postmatch.
"We needed two centre-halves ideally. With Wesley going we needed a replacement for him. Wout is an experienced player, a leader. He will come in and give us that personality. We are delighted he has joined us."
Information from Reuters was used in this report.