The result comes ahead of the Red Devils' blockbuster clash with champion Liverpool at Anfield on Monday (AEDT).
One moment of magic from James Maddison made the difference against Southampton as he smashed high into the roof of the net from a narrow angle on 37 minutes.
The midfielder also remembered to adhere to new Premier League guidelines on goal celebrations as he mimicked handshakes and high-fives at a distance.
"If little things like that are required to keep football going that's what we need to do," Maddison said after criticism from politicians over exuberant celebrations while infection rates of coronavirus in England continue to soar.
"Footballers get a lot of stick. It's nice to turn something like that into a lighter note because the whole country wants football to continue."
Stuart Armstrong struck the bar for Saints in the second-half, but the visitors badly missed the presence of top scorer Danny Ings up front, who was absent because of a positive COVID-19 test, and the Saints missed the chance to leapfrog Chelsea and Tottenham Hotspur on the table.
Harvey Barnes sealed the points for Leicester deep into stoppage-time on the counter-attack, putting the Foxes in buoyant mood ahead of their upcoming league clash against Chelsea.