Oli McBurnie and Anel Ahmedhodzic appeared to have done enough to propel the Blades to victory after Alfie Doughty's first-half effort, but Chris Wilder's men masterminded their own downfall, as Jack Robinson and Anis Ben Slimane both turned into their own net to gift Luton back-to-back top-flight wins.
On what was a momentous occasion for referee Sam Allison - the Premier League's first black referee in 15 years - Ryan Giles replaced the injured Issa Kabore for Luton, while Blades boss Wilder tinkered with his system by bringing in Gustavo Hamer, James McAtee, McBurnie and Ahmedhodzic.
Riding the crest of the wave from their beating of Newcastle, Luton pinned Sheffield United back inside the opening 10 minutes - forcing a couple of their favoured corners - but Wes Foderingham was not given anything to think about.
However, the Sheffield United number one may have wanted the Bramall Lane turf to swallow him whole in the 17th minute, as Luton took a deserved lead thanks to a horrendous piece of goalkeeping from Foderingham.
The dangerous Doughty picked up the ball on the right-hand side from Albert Sambi Lokonga, gave Hamer the slip and ostensibly tried to square across the six-yard box, but his cross-cum-shot squirmed through Foderingham's legs at his near post and nestled into the back of the net.
Doughty's goal belatedly sparked Sheffield United into life, as Hamer forced Thomas Kaminski into a comfortable save from a free kick in the 30th minute, before McAtee's long-range curler bounced just wide three moments later.
However, the visitors remained a constant threat from dead-ball situations, and Doughty almost turned provider for Ross Barkley in the 38th minute, but his header from the 24-year-old's corner was a fraction too high.
In the first and only minute of injury time, Sheffield United's best chance of the half coincidentally arrived in similar circumstances, as Auston Trusty made a brilliant run to the near post to meet Hamer's corner but could only clip the top of the crossbar.
The Hatters' half-time advantage was a fully merited one, though, and Wilder responded at the break by introducing Max Lowe for Andre Brooks on the left-hand side.
The game opened up as Sheffield United began to make inroads in Luton's defensive third, and after the Hatters spurned a couple of half-chances to add to their advantage, Bramall Lane erupted in elation in the 61st minute.
A smart link-up between McAtee and Cameron Archer ended with the former driving into the box and being scythed down by Sambi Lokonga, but while on the ground, he instinctively flicked the ball into the path of McBurnie, who did the rest with a calm finish into the far corner.
Just eight minutes later, the unforeseen Blades comeback was seemingly complete, as following an almighty goalmouth scramble - in which Archer was thwarted twice in quick succession by Lokonga and Teden Mengi - the ball fell for Ahmedhodzic to fire home into the side of the net.
However, another eight minutes passed a red and white shirt undid all of his side's hard work, as Robinson sought to prevent Gabriel Osho from connecting with a Carlton Morris cross, but in doing so, he could only guide the ball into his own net with an inadvertent flick of the head.
Christmas Day may have been and gone, but Sheffield United were still in a giving mood, as with 81 minutes on the board, another Morris cross cannoned off the thigh of Ben Slimane, looped over Foderingham and nestled into the hosts' goal.
Not even 10 minutes of second-half added time was enough for Sheffield United to save their bacon, and their self-inflicted downfall means that they remain bottom of the pile, while Rob Edwards's team are still below the dotted line in 18th but are only one point adrift of safety.
Sheffield United now prepare to close out the year with the most difficult test possible, a trip to Manchester City on Saturday, while Luton host a struggling Chelsea in the lunchtime kickoff on December 30.