Blades boss Neil Warnock's mood is also likely to be decidedly frosty as his side wasted the opportunity to leapfrog West Brom into the automatic promotion places.
Warnock can have no complaints either as Tudgay's fine header after 70 minutes was just reward for an enigmatic start to the second half.
Midfielder Mark Rankine went close for the Blades with a fine header but as the temperature plummeted, so did United's chance of rescuing a point as substitute McLeod swept the ball home with the last kick of the game.
And for manager George Burley, a crucial victory was all the more memorable as it is fierce rivals Nottingham Forest who replace the Rams in the bottom three.
However, United enjoyed the lion's share of possession in the opening exchanges as Nick Montgomery and Paul Shaw both went close with speculative long-range strikes.
However, the home side slowly dragged themselves into the contest and carved out a superb opportunity to open the scoring after 10 minutes.
Candido Costa's superb delivery into the penalty area found Marco Reich, who could only scuff his strike into the grateful hands of Paddy Kenny when it seemed easier to score.
The visitors were not playing well, but such is the confidence within their ranks - United have lost just once in 10 games - they almost took the lead just before the half-hour mark.
Alan Wright sent over a dangerous ball into the area from the left flank which found Rankine, whose bullet header was goalbound - but expertly dealt with by Lee Grant in the County goal.
Derby began the second half with purpose and with 55 minutes gone, Burley's men had legitimate claims for a penalty when debutant Leon Osman's mazy run was brought to an abrupt end by Robert Page's arm.
The Everton loan signing went down a little too easy though, and this may have prompted referee David Crick to wave away County's vehement appeals.
United failed to see the warning signs and were punished after 70 minutes through Tudgay's fine finish.
Reich made amends for his first-half miss with a delightful ball to the 20-year-old, who had time and space to head home from six yards.
United went about restoring parity but Derby's stout rearguard was in no mood for compromise and kept the Blades' stagnant frontline expertly at arm's length.
And a fine performance was galvanised at the death when United failed to clear their lines and McLeod - who replaced Reich after 85 minutes - was on hand to stab the ball into an empty net.