But United boss Chris Wilder was disappointed not to have claimed all three after a stirring display, especially in the second half, against the second-placed hosts.
Matej Vydra had earlier put the Rams in front from the penalty spot midway through the first half of the Sky Bet Championship clash and both teams threatened to win it on the break several times in the closing stages.
The visitors made the better start, with Richard Keogh blocking an early shot by Clayton Donaldson.
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United got into some dangerous positions without finding the ball to open up Derby's defence and, after Mark Duffy was booked for catching Johnny Russell, the home side went in front in the 24th minute.
Russell received treatment but was involved in the goal when he took the ball off George Baldock near the goal-line and, when the defender slid in to win it back, he appeared to handle the ball. Referee Darren Bond immediately pointed to the spot and Vydra smashed the penalty to the left of Simon Moore and into the roof of the net.
Russell was just off target with a 20-yard effort as Derby's quick movement up front threatened to expose United, but the visitors almost equalised in the 39th minute when Donaldson broke down the left and forced Scott Carson into a diving save at his near post.
Derby responded with a swift counter-attack that ended with Andreas Weimann curling a shot just wide, but the Blades were still in the game and went close again in the 55th minute.
Enda Stevens cut the ball back to John Fleck on the edge of the area and his low shot rolled just wide of Carson's left post but, two minutes later, United were level.
Donaldson swung the ball in from the right and Clarke was unmarked six yards out to steer a header past Carson.
Clarke was nearly in again when Derby gave the ball away in a central area, but United were nearly caught on the break when Russell's pass almost put David Nugent through.
Both teams gave the ball away as rain began to fall and Clarke was close to playing in Donaldson, who had worked tirelessly up front, but both defences stayed strong and neither side could fashion an opening in the final minutes.