A fifth consecutive win for Boro was secured thanks to first-half strikes from Tommy Conway and Riley McGree as they climbed ahead of Coventry, despite a nervy finish after Patrick Bamford's second-half goal.
Frank Lampard's Coventry were held to a goalless draw at home to Oxford on Saturday, opening the door for Kim Hellberg's side.
They raced out of the blocks at Bramall Lane as Conway finished unerringly across Blades goalkeeper Michael Cooper with less than 20 minutes on the clock.
McGree then doubled the visitors' advantage on the stroke of half-time as he reacted quickly to nod into an unguarded net after Hayden Hackney's effort from distance hit the post and rebounded towards him.
Boro refused to sit on their laurels in the second period and tried to secure a third goal, which would have effectively ended the match as a contest.
But it was Sheffield United who struck next as Bamford supplied the finishing touch in the 73rd minute at the end of fine play by Kalvin Phillips and Gus Hamer.
It was now a matter of seeing out the match for Boro, and that task was made easier when Joe Rothwell, on for his Blades debut, was given his marching orders for a poor challenge on Alan Browne with 10 minutes remaining.
