The right-back, who came through the ranks at the Blades, struck midway through the first half with a superb long-range strike.
Ramsdale did well again in the second half when he dived at full stretch to keep out a Riyad Mahrez free-kick.
But from then on the visitors struggled to find a second goal that would kill the game off and United should have capitalised when substitute John Lundstram fired just over from Sander Berge's cross.
That was as close as the Blades came as City took their third league win of the season to move into the top half of the table.
Sheffield United remain winless and in the relegation zone with just one point from their first seven games.