Jamie Vardy scored the only goal of the game at the King Power Stadium, as his looping header over goalkeeper Jamal Blackman in the 66th minute settled a hard-fought tie.
Riyad Mahrez provided the assist on his first start for the Premier League side since January 20, the winger's deep cross setting up his unmarked team-mate at the back post.
Yet the hosts needed a moment of excellence from Kasper Schmiechel to deny Sheffield United an equaliser, the goalkeeper stretching low to his left to keep out a first-time effort from George Baldock that looked destined to find the corner of the goal.
Instead, Vardy's clever finish proved enough to send Claude Puel's team through to the last eight of the competition.
The England international had wasted a glorious chance to put the hosts ahead in the closing stages of the first half, shooting straight at the advancing Blackman after being put through on goal by the ebullient Mahrez.
Blackman also kept out a long-range drive from Onyinye Ndidi, while at the other end a familiar face prevented the Championship side from going ahead before the break.
Harry Maguire - who came through the ranks at Bramall Lane before moving to Hull City in 2014 - produced a brilliant block to divert Enda Stevens' goal-bound effort over the top of the net.