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.
Major moment
While Jamie Vardy’s fifth goal in as many games for City was of course vital to the end result, Kasper Schmeichel’s save with 18 minutes to play was also huge.
The Denmark international got a strong hand to George Baldock’s low effort to help the Foxes secure a spot in the last eight of the competition.
Who impressed?
“He’s been the best passer of the ball by far tonight.” – Nev Foulger, LCFC Radio
“He created umpteen opportunities in the first half. He’s had a really good game.” – Gerry Taggart, LCFC Radio
Riyad Mahrez’s display against Sheffield United earned plenty of plaudits from the LCFC Radio summarisers on Friday night.
In the first period, the Algerian created numerous goalscoring opportunities, while his pass for Jamie Vardy’s header after the break was simply excellent.
Where do we stand?
The Foxes secured their spot in the quarter-finals with the triumph and will discover their opponents on Saturday evening at around 8pm.
Coming up…
Next up for City is a Premier League encounter with Stoke City at King Power Stadium on Saturday 24 February (12:30pm kick-off).
For full match statistics, click here.