Gylfi Sigurdsson struck in the 80th minute as the Toffees have pulled to within three points of arch-rival Liverpool for top spot in the table.
Sheffield United remain winless on the season, rooted to the bottom of the table, ten points from safety through 15 games.
Story of the match
Sheffield United came out of the blocks well, but it was Everton who had the first chance in the 11th minute as Sigurdsson swirled in a corner that Aaron Ramsdale was able to collect.
Five minutes later, the hosts almost took the lead. Rhian Brewster played David McGoldrick in before he let a tame shot go that would have been ruled out as he was judged to be offsides.
Alex Iwobi then tested Ramsdale, as he snapped off a volley that the Sheffield United goalkeeper collected. Dominic Calvert-Lewin had the Toffees' next chance, curling an effort just wide of the post on the half hour mark.
Ramsdale was almost caught out a minute later, spilling Sigurdsson's free kick and watching the ball fall to Ben Godfrey, who smashed one right into the keeper's midriff.
Iwobi was creating, and two minutes after his efforts were denied, Sigurdsson snapped a shot off that was set up by Anthony Gordon, but flashed just wide.
Just before halftime, the hosts came close again. A throw-in by Ben Osborn was collected by Oliver Burke, down the left channel, his shot however, went inches wide.
Everton continued to press and in the 56th minute, Iwobi slotted in Calvert-Lewin who tried to muscle out John Egan, but the Blades' defender is first to the ball and saw off the danger.
Sheffield United appealed for a penalty when Oli McBurnie was taken down by Michael Keane, but VAR ruled a slight push, but not enough to warrant a spot kick.
Everton continued to have chances through Mason Holgate, Calvert-Lewin and Sigurdsson. Until, ten minutes from time, the Icelandic front man finished past Ramsdale in the bottom corner.
McBurnie had two chances in stoppage time to rescue a point for the Blades, but failed to put his efforts on target, summing up what has been a historically bad season in Sheffield so far.
Man of the match: Gylfi Sigurdsson
In a match that was, at times, short on quality, Sigurdsson was there to calmly slot home the winner. The conditions were terrible, but he still managed to finish as well as helping to create other chances for Everton.
The buildup to the goal showed why Everton are surprise title contenders
Seamus Coleman played the ball into the box for Calvert-Lewin and after it was headed clear by Sheffield United, Bernard needed just one touch to play it square to Abdoulaye Doucouré, who got it to Sigurdsson for the finish past Ramsdale. A thing of beauty.
Sheffield United played well, but look ahead
A point was all the Blades could have really expected. It looks for all the world, barring the greatest escape in Premier League history, that they'll be back in the Championship next year, but if they want to have even the slightest chance of staying up, performances like this against Burnley and Crystal Palace in their next two fixtures are a must.