The Blades are still searching for their first win in 12 games under new boss Micky Adams and are now six points from safety in the table.
The home side started brightly and almost made their early domination count in the 14th minute but Rams defender John Brayford nodded Neil Collins' goalbound header off the line.
Michael Doyle was next to go close for the Bramall Lane side but his header from six yards flew over the bar following Bjorn Helge Riise's 18th minute corner.
Derby took the lead against the run of play in the 27th minute through Robinson's first goal for the club following his arrival on loan from Millwall last week.
Daniel Ayala's left wing cross was not cleared properly and it fell to Robinson, who was unmarked on the edge of the six-yard box, and he tapped the ball into the back of the net.
Chris Porter went close to doubling his side's advantage in the 35th minute but his deflected shot from 20 yards flew just inches wide of Steve Simonsen's post.
Blades top scorer Ched Evans should have equalised just two minutes later but he put his effort from 12 yards well over the bar before he fired a 43rd minute effort straight at on-loan Derby keeper Frank Fielding.
The home side started the second half the same as they started the first - with plenty of possession, but failed to make it count despite a number of early openings.
Riise stung the palms of Fielding just 20 seconds after the restart with a fierce drive from the left channel before Matthew Lowton and Evans failed to trouble the Rams keeper with long range efforts.
Lee Williamson also headed another Riise corner over the bar before Richard Cresswell missed United's best chance of the game in the 49th minute, when he headed Nyron Nosworthy's cross over the bar when well positioned.
Derby did have chances of their own, most notably Robinson's long range effort which Simonsen did well to tip round the post, before the Blades surged forward again in search of an equaliser.
Adams introduced Marcus Bent in the 70th minute and the switch almost paid dividends but Lowton headed Bent's right wing cross straight at Fielding before Evans missed another good chance and Riise's effort from the rebound was blocked at the near post.