Clough, who followed in late father Brian's footsteps when appointed Derby boss in 2009, said his dismissal at the end of last month had left a bitter aftertaste and had taken him and coaches Gary Crosby and Andy Garner by complete surprise.
The 47-year-old, who spent nearly five years in charge at Pride Park, was installed as David Weir's successor at Bramall Lane on Wednesday and signed a two-and-a-half year contract to become the Blades' eighth permanent manager in the last six-and-a-half years.
"It was fairly shocking, yes," said Clough, who has been joined at Bramall Lane by both Crosby and Garner.
"I don't think we're over it, that's why this will certainly help in that way.
"We gave five years of extremely hard work, but I think it also helped in coming back in straight away.
"Quite a few managers have a rough time and say they need a break, which we weren't averse to.
"But we didn't have a rough time in the last season or two. We thought we were doing well.
"We had a great team on the pitch, a young team playing some great football, so from that point of view we didn't really need a break. We've still got lots of energy.
"We didn't really have much of an inkling about it, all the noises at the time were quite positive at the club about the job we had done over four or five years.
"To be told all of a sudden that it's not good enough surprises you somewhat.
"Generally in football nothing surprises you, but this one did.
"Everything that happened. We're obviously going to have to concentrate on not dwelling on it because we're going to have everything on here, looking forward now.
"We could have wallowed for six months, waiting for a so-called bigger job, but we're great believers in if you get knocked down you get back up again and get on with it.
"I certainly think we can use it in the short-term as additional motivation.
"I think we're reasonably motivated as it is, but in the short-term we'll certainly be using it.
"There's certain feelings that are still there that we'll try and keep beneath the surface for as long as we can."
Clough steered Derby to Championship safety on the penultimate day of the 2008/9 campaign and the club finished 14th, 19th, 12th and 10th in the following four seasons.
He said the Blades had huge potential, while the club's infrastructure and fan-base were key factors in opting to drop down a division.
"I don't think you can pick and choose in football," he added. "I know jobs come up in the Championship on average every two years, but are they going to be better than this one?
"Are they going to be better clubs than this one, better people to work with? You never know.
"They made it clear they wanted us to come here and do the job and that's a big part of it."
Clough will take charge of his first game as Blades boss at Bramall Lane against Crewe on Saturday.