Pitman broke his duck with his first league goal for the Robins in the defeat of Leicester last weekend and he added two more to his account with spot-kicks after six and 34 minutes.
United's day got worse when Richard Cresswell was sent off moments before half-time.
Their misery was compounded just after the hour mark when Jamie McAllister's corner flew straight over goalkeeper Steve Simonsen and into the net.
Keith Millen's side were given the opportunity to take an early lead when Leon Britton brought down David Clarkson. Pitman seized the chance with a superbly-placed penalty into the top corner.
United responded positively and Cresswell and Matt Lowton had efforts on goal but it was easy work for David James.
Albert Adomah threatened in the 30th minute before a handball by Stephen Jordan enabled City to extend their lead. Pitman obliged again from the spot.
Simonsen kept the Blades in the game with a block to deny Clarkson and Cole Skuse shot over the crossbar.
United's task became more difficult when Cresswell went charging in on Marvin Elliott. Referee Pat Miller did not hesitate in showing the striker a straight red card.
It took the hosts a while to press home their numerical advantage, with Adomah shooting just wide before the hour mark. Simonsen then saved from Pitman.
However, the keeper then misjudged McAllister's inswinging corner and City had a three-goal lead.
Clarkson sent a shot wide in the 79th minute before a block by Kyle Bartley denied Pitman the chance of a hat-trick.
Clarkson continued to try to get his name on the scoresheet but he was thwarted by Simonsen once again with two minutes remaining.
Ched Evans came close to a consolation goal for United but his long-range effort struck the crossbar in the first minute of injury time.
Steven Caulker had the final chance as he shot across goal in the dying moments.