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.
Source: PA
Source: PA