Paddy Kenny denied Palace teenager Sean Scannell before Julian Speroni saved a Rob Hulse shot with his legs and then Jon Stead fired well wide when through.
Scowcroft netted on 38 minutes from Ben Watson's free-kick and Tom Soares went close to making it 2-0 after the break.
The Blades dominated after that but Speroni again thwarted Hulse, who also shot wide from two yards out.
Sheff Utd boss Bryan Robson:
"I don't think it's anything to do with a lack of confidence - we just haven't been getting the breaks.
"If we continue to create the amount of chances we're creating, they'll start going in at some stage."
Crystal Palace boss Neil Warnock:
"We are limited but the lads gave their all and that's all I can ask for. We're not the finished article but we've moulded a unit and worked hard and we're getting results.
"We will slip up but I think anyone can get into the play-offs this season."
Sheff Utd: Kenny, Bardsley, Kilgallon, Lucketti, Naysmith, Gillespie (Sharp 53), Tonge, Montgomery, Armstrong (Hendrie 63), Hulse, Stead.
Subs Not Used: Bromby, Bennett, Carney.
Booked: Hendrie.
Crystal Palace: Speroni, Butterfield, Lawrence, Hudson, Hill, Watson, Soares (Ifill 90), Derry, Scowcroft, Morrison (Fletcher 73), Scannell (Hills 78).
Subs Not Used: Freedman, Fonte.
Booked: Derry, Soares, Butterfield.
Goals: Scowcroft 38.
Att: 23,982
Ref: Andy D'Urso (Essex).
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