Hart was appointed as the Eagles' new boss this week following Neil Warnock's switch to London rivals QPR and the former Portsmouth manager saw his reign get off to the best possible start as Alan Lee grabbed the only goal of the game midway through the first half.
United are looking to secure a play-off place but they barely threatened Eagles goalkeeper Julian Speroni on their way to a sixth successive away defeat.
Lee put Palace ahead in the 23rd minute as he coolly volleyed past Mark Bunn from Danny Butterfield's superb right-wing cross.
The hosts created the better opportunities thereafter but were unable to put the game to bed with a second goal.
Yet it mattered not as they held on to their slender advantage with relative ease in the closing stages.
Hart made two changes to the Palace side that drew 1-1 with Doncaster in Warnock's final game in charge, with Johannes Ertl and Calvin Andrew starting in place of Matt Lawrence and Kieran Djilali.
Stephen Quinn replaced Henri Camara in the only alteration to the United line-up.
Neither goalkeeper was troubled early on as both sides struggled for fluency in blustery conditions.
But when a clear-cut chance did finally arrive, it was clinically dispatched by Palace striker Lee.
Butterfield floated over a delightful cross from the right and Lee - making his 300th league start - was on hand to slot home a right-footed finish from close range.
Andrew had an opportunity to extend the Eagles' lead soon after, but the young forward fired wildly off target from a promising position.
United were struggling to offer any attacking threat and the home side looked increasingly comfortable as the half wore on, with the powerful Andrew causing all sorts of problems.
The Blades eventually called Speroni into action in the 41st minute as Jamie Ward diverted a Mark Yeates free-kick into the arms of the Palace goalkeeper.
Lee and Kyle Bartley were forced off shortly before the break following a clash of heads, but both men were able to continue following treatment.
United picked up the pace at the start of the second half, but Palace came close to scoring again in the 53rd minute as Darren Ambrose tested Bunn from 12 yards.
Lee then fired over from six yards following good work from Nick Carle on the left and the impressive Andrew also saw another chance go begging as he missed the target with a header from Ambrose's cross.
The visitors huffed and puffed late on in search of an equaliser but Speroni remained relatively untroubled in the Palace goal.
And Ambrose almost added a second for Palace when he saw a free-kick parried clear by Bunn.