Clarke opened the scoring after eight minutes, collecting the ball 30 yards out before unleashing a fine effort into the top corner which gave Frank Fielding no chance.
The Blades doubled their lead on the half-hour mark as Sharp connected perfectly with a right-footed volley, the ball striking the underside of the crossbar before crossing the line.
City were all over the place defensively and it was no surprise when United went three up, Freeman was given too much space before volleying home.
Aden Flint gave the home side hope when he converted Josh Brownhill’s left-wing corner with a firm downward header.
And City reduced the deficit to a single goal with 15 minutes remaining, Joe Bryan netting with a superb left-footed free-kick from just outside the box.
But despite a rousing finish, City paid dearly for their first-half showing as their opponents moved above them by one place to finish 10th in the Championship table.
Jamie Paterson headed over for City from a Bryan cross before top-scorer Bobby Reid wasted their best chance of the half, volleying wide from Korey Smith’s chip into the box.
But, in truth, the Robins had been poor and the home fans vented their anger with loud booing at half-time.
Head coach Lee Johnson responded by sending on centre-back Hordur Magnusson for Paterson at the start of the second half, moving Lloyd Kelly to left-back and Bryan forward.
They upped their game and Flint’s headed goal gave them a timely lift, and by the time Bryan netted they were on top.
But United remained dangerous on the break and although they were unable to add to their tally, Chris Wilder’s men looked comfortable at the back as they held on.