The Blades have struggled away from Bramall Lane this season and manager Kevin Blackwell knows they will have to sort out their travels if they are to maintain a push for the promotion and play-off places.
Crystal Palace, a side in administration and struggling at the foot of the table due to having points deducted, performed well for new manager Paul Hart.
Blades boss Blackwell said: "I am frustrated, but it's always tough facing a new manager and a new team and I thought they fought like dogs.
"In the first half they battled well and in the second half, although we dominated, we just didn't create enough."
Hart praised his players after they stayed out of the relegation zone thanks to the win.
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The cash-strapped Eagles had endured another traumatic week following the departure of Neil Warnock to Championship rivals QPR.
Hart has been charged with trying to save Palace from the drop and kicked off his tenure with a priceless victory thanks to Alan Lee's 23rd-minute effort.
He said: "From the minute I came in, the way the players behaved and spoke told me they are a strong bunch and they went out there and proved it.
"I was absolutely delighted for them. I think the performance was excellent. All over the pitch we did what was necessary.
"Sheffield United, whether they are in the Premier League or the Championship, are a very tough team and you always know you have been in a game against them."
Palace are only fighting a relegation battle after being docked 10 points for going into administration and were well worth the points as they dominated for virtually the whole match.
They only looked like surrendering their lead in the final seconds when United eventually showed some attacking intent.
Lee's goal came midway through the first half when the former Republic of Ireland striker stole in ahead of Chris Morgan to fire home from Danny Butterfield's cross.
Darren Ambrose spurned a good chance after the break, while Lee missed a great opportunity to grab his eighth goal of the season moments later, nodding Nick Carle's cross over the bar ,but the Blades rarely threatened at the other end.
Crystal Palace: Speroni; Clyne, Butterfield, Davis, Hill; Ambrose (Hills 90), Derry, Carle (Scannell 90), Ertl; Andrew (N'Diaye 87), Lee. Unused substitutes: Manns, John, Lawrence, Djilali.
Sheffield United: Bunn; Nosworthy, Morgan, Bartley, Kallio; Yeates (Evans 78), Montgomery, Williamson (Harper 70), Quinn (Camara 58); Cresswell, Ward. Unused substitutes: Bennett, Taylor, Stewart, Seip.
Referee: Andy Penn (W Midlands).
Man of the match: Darren Ambrose.
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