The Toffees had gone ahead with an early Mikel Arteta header, but when Claude Davis was shown the red card after Johnson sprawled in the box, James Beattie's penalty should have ended the game as a contest.
But the Blades - far from impressed with the penalty award - responded with a commendable second-half fightback.
The visitors threw men forward and made Everton look ordinary - but their efforts ultimately proved fruitless.
Unless Blades boss Neil Warnock finds someone who can score goals at this level - they have only managed four in nine games - their stay in the top flight will be brief.
Everton were forced to use central defender Joleon Lescott at left-back and midfielder Phil Neville on the right side of defence due to the injuries to Tony Hibbert and Nuno Valente, while veteran David Weir made his first start of the season.
Simon Davies was given another chance on the right flank while Beattie came in up front alongside Johnson.
The former Crystal Palace striker's pace and tenacity twice threatened to embarrass Davis and Phil Jagielka before Everton took a 13th-minute lead.
Davies' determined run across midfield ended with a ball to Neville, who fired over a cross despite being fouled by Leigh Bromby.
The linesman flagged, but referee Dermot Gallagher waited just a few seconds to see where the move would end - and it proved a good move as Arteta headed low into the corner of the goal.
Beattie saw two efforts go close, Davis pressuring him into a hasty effort - and the next time Davis applied some pressure, it ended with a red card and a penalty.
In the 33rd minute Davies and Arteta combined to send Johnson racing away, with Davis in pursuit.
The England forward went crashing in the box, although TV replays suggested only minimal contact.
But it was enough for Gallagher to instantly point to the spot and dismiss Davis, before Beattie blasted in the spot-kick.
The Blades immediately sent on defender Chris Morgan to fill the gap left by Davis - front man Alan Quinn being sacrificed - before wasting two efforts to get back into the game.
First Colin Kazim-Richards deflected a Rob Hulse header goalwards, with Tim Howard making an instinctive block to keep out the effort.
A minute later Hulse got away on the left and pulled an effort across goal and inches wide of the far post.
But there was still time for Davies to set up Johnson, who drilled a shot against the woodwork.
The Blades started the second period with commendable spirit, and should have scored when Tim Cahill's misplaced pass allowed Chris Armstrong to send Leigertwood clear - but his fierce cross-shot flashed inches wide of the far post.
Everton sent on James McFadden and Leon Osman for Beattie and Tim Cahill respectively, with the Blades responding by taking off Kazim-Richards and sending on Danny Webber.
Six minutes later Steven Kabba replaced Keith Gillespie as the Blades' three forwards kept Everton's defenders occupied.
Osman wasted a good opening when Neville's pass put him clear, but Everton were being asked to defend and Hulse rose to head just over from a free-kick.
Michael Tonge was booked for a foul on Arteta, while a minute later Everton sent on Victor Anichebe for Neville, Davies reverting to right-back.
There had been very little to trouble Paddy Kenny in the second period, and when McFadden's 18-yard effort sailed wide with 10 minutes left, Everton were no nearer their third goal.
Webber then saw a close-range effort cannon back off Howard's shins, with the Blades hardly looking like a 10-man team.
Anichebe could have found the net with a 60-yard run and shot over, before flashing another effort wide.
But when McFadden's 88th-minute header produced the first genuine save for Kenny in the half, it spoke volumes about Everton's efforts after the break.