Two first half goals came from Shaun Murphy and Jon-Paul McGovern and two second half strikes from Steve Kabba took the Blades through to round four.
The Blades took the lead in the ninth minute with their first attack, McGovern's corner from the right hand side was scrambled off the line after Murphy headed but Shaun Murphy put in the rebound at the second attempt.
Paul Peschisolido was let loose on the left flank four minutes later where his low cross met Nick Montgomery where his shot was well saved by Robins goalkeeper Steve Book.
Cheltenham then had a change to level the score line when right-back Neil Howarth played a one-two with Richard Forsyth who rushed into the box but Murphy blocked as he tried to beat him.
A minute later the Blades doubled their lead, Steve Kabba put a low cross from the left hand side, Peschisolido swung at the ball but missed at the near post, but Jon-Paul McGovern steamed in late behind the Canadian International and smashed the ball past Book from six yards out.
United where in charge of the game and went close to a 3rd when Jon-Paul McGovern's right-wing cross came off Michael Duff's head and fell to Michael Tonge, but his shot which was going wide forced a corner from Book.
In the 41st minute Book was called into action yet again when Jagielka shot past him after Peschisolido teed off a short pass to the right back.
On hour mark Jagielka had another pop at goal but this one was much better, his right footer shot from20 yards out brought out a fantastic save from Book as he dived low to his right.
However Book could not stop the Blades scoring their third in the 63rd minute as they killed the game. Steve Kabba partnered up with Peschisolido where Peschisolido cross low to Kabba who fired low across Book into the far corner.
Kabba drilled another low shot wide from 15 yards five minutes later and almost scored with a deflection off his backside with 15 minutes remaining after McGovern's shot hit him and almost deceived Book.
The Blades almost added a fourth in the 78th minute when substitute Laurens Ten Heuvel, on for Peschisolido, beat John Brough and rolled the ball across to Montgomery, whose powerful shot from eight yards was well tipped over by Book.
Howarth headed over the top from the right-wing corner with eight minutes remaining.
The fourth did come for the Blades in the 84th minute when Kabba scored from close range with a back-header following a spell of head tennis as Cheltenham failed to clear their line from a corner.
Cheltenham were then denied an injury-time consolation goal when substitute Julian Alsop's cut-back from the right was cut out by Kenny with another substitute, Paul Brayson, waiting to pounce.