Making six changes to his starting XI, Claude Puel deployed a 4-4-2 stratagem, with skipper Wes Morgan making a return from a hamstring injury.
The Foxes started well, with Blades captain Jake Wright blocking a Vardy effort before the England international curled just inches wide shortly after.
Wilfred Ndidi then forced Jamal Blackman into a good save following a powerful long-range strike, as City continued their search for opener.
Former Sheffield United defender Harry Maguire put in a brilliant block to prevent Enda Stevens from firing his side in front, while Vardy saw a shot saved just moments later by Blackman.
After a goalless first half, City continued to threaten and their persistence paid off on 66 minutes, when Vardy got up well at the back post.
Riyad Mahrez sent in a delightful delivery and Vardy was on hand to nod over Blackman and into the bottom corner of the net for 1-0.
Demarai Gray, lively throughout, saw a curling effort saved by Blackman, while at the other end, Kasper Schmeichel was on hand to pull off a vital stop to prevent George Baldock from netting an equaliser.
Further attempts from Samir Carruthers, Ched Evans and Mark Duffy followed, but the Foxes stood firm late on to secure the triumph.
Hasta 6 cambios con relacion al equipo inicial de las ultimas jornadas, Paul Claude volvio a su formato de 4-4-2, con la reaparicion en la ultima linea de Wes Morgan. haciendo pareja con Harry Maguire.
Wilfred Ndidi tuvo las mas clara exigiendo al portero Blackman, mientras que Maguire tuvo una barrada espectacular para evitar la apertida en esfuerzo de Stevens.
Despues de un primer tiempo sin goles, Leicester City encontro respuesta a su propuesta con una genial jugada de Marhez al area como con la mano y una mejor definicion de cabeza de Jamie Vardy para el 1-0.