United appeared poised to pull away from the relegation zone when Rob Hulse scored his eighth goal of the season in the 52nd minute.
But 15 minutes from time a mistake from Kenny enabled Mikel Arteta to rifle home the equaliser from the spot - the fifth penalty this season given away by United to Liverpool and Everton.
The first came on the opening day of the season against the Reds in a 1-1 draw, followed by another in a 2-0 defeat at Goodison Park in October.
Last week they conceded two spot-kicks early on as they suffered a 4-0 defeat at Anfield, so it was no surprise when their penalty problems continued.
The Blades will be hoping this one does not prove too costly as they remain seven points clear of the relegation zone with nine matches remaining.
In the end, a draw was arguably a fair result as the game was a tight, nip-and-tuck affair between one side gunning for a place in Europe, and another eager to retain their Premiership status.
The best chance of a relatively sterile opening half fell the way of Tim Cahill with a downward header from Gary Naysmith's cross.
Although Kenny was easily beaten, returning centre-back Claude Davis was on hand to hack the ball off the line in his first game since January 20.
It was a reshuffled rearguard that primarily kept Kenny out of trouble, with Davis partnered by captain Chris Morgan as he started for the first time since January 1, allowing Neil Warnock to push Phil Jagielka back into midfield.
Kenny, though, was almost the architect of his own downfall in the 10th minute, squinting into the bright sunshine as he pawed at a bouncing throw from Toffees captain Phil Neville that almost caught him unawares.
Fortunately for Kenny, Derek Geary and Davis were on hand to clear the danger before the duo then became involved in two of the three elbowing incidents that dominated the half.
Davis was caught by James Beattie as they jumped for an aerial ball, the latter earning a caution from Mike Riley that might not have been given on another day.
But then Stephen Quinn was clearly flattened by a forearm from Cahill that somehow went unpunished, before Arteta's leap left Geary with a bloodied lip.
United were restricted to just two serious chances, both to Jon Stead who forced comfortable, low saves out of Tim Howard on each occasion.
But the fans did not have to wait too long in the second half for the game to rise from its lethargy.
Seven minutes in, Hulse stole in front of the covering Joseph Yobo to steer a right-wing cross from Colin Kazim-Richards past Howard.
Hulse had been anonymous before the break, but appeared to let Kazim-Richards know that he had been waiting for such deliveries when the two met to celebrate.
Warnock's side then looked comfortable until Kenny's gaffe cost his side victory.
Jumping to collect a high ball, he fumbled, allowing Johnson to steal in.
The England international may have a reputation for going to ground too easily, but that was not the case on this occasion as Kenny caught his ankle as he attempted to retrieve the ball.
After Riley pointed to the spot, Arteta then drove the ball home off the underside of the crossbar before almost netting the winner two minutes later.
Kenny partly redeemed himself for his earlier, indiscretion with a superb save as a 22-yard free-kick from Arteta whistled through a crowd of players.
There was further angst for Warnock, though, in the closing stages when Morgan went off with what appeared to be a broken nose after accidentally heading the back of Everton substitute James Vaughan's head.