A Brian Howard strike in the first half was enough to secure three points, after the home side missed two golden opportunities.
Nathan Tyson and Robert Earnshaw both failed to convert after being sent clean through - leaving Forest stranded in the bottom three and Calderwood to face more speculation about his future.
It took a fine save from Lee Camp to deny United an early goal, as the keeper dropped acrobatically to his left to swat away a James Beattie header, with Greg Halford then firing the loose ball badly wide.
Joe Garner had Blades keeper Paddy Kenny in a panic with a glancing header that dropped narrowly wide.
But Kenny was up to the task when Chris Cohen sent Earnshaw racing clear with an incisive pass, with the keeper spreading himself well to make a vital save - with Paul Anderson firing the rebound wildly over.
It was the visitors who edged their way into the lead in the 31st minute, as Forest failed to clear a corner.
Kelvin Wilson had done well to beat Beattie in the air, but saw his attempted clearance fall to Howard, who lashed a volley inside the post from eight yards.
Forest should have pulled level before half-time, as Anderson's vision sent Tyson scampering through unmarked, only for the striker to prod a weak effort wide.
After the break, chances were at a premium as United comfortably defended their lead and looked to hit Forest on the break.
Garner dived bravely to connect with a header at the near post, but sent his effort wide, while Kenny had to palm over in desperate fashion after an attempted clearance was charged down by Tyson and threatened to loop in.
While Tyson had Kenny at full stretch again, this time with an arcing header that the keeper again palmed over.
But that was as close as Forest got to an equaliser, as United held firm.