Brayford, on loan from Cardiff until the end of the season, struck the decisive second goal in Sunday's 2-0 quarter-final triumph over Charlton at a sold-out Bramall Lane.
The Blades became the first club from the third tier to reach the last four of the FA Cup since Wycombe in 2001 and extended their winning sequence to nine matches in all competitions.
"It's basically down to the lads, the gaffer won't take any credit for it himself," Brayford said.
"But the way he's gone about telling the lads what he wants to do, it speaks for itself really."
Clough, who replaced David Weir as boss at the end of October following the club's disastrous start to the season, steadfastly refuses to take any credit for a remarkable turnaround.
His side have climbed out of the bottom three and into the top half of Sky Bet League One while securing eye-catching cup wins over Aston Villa, Fulham, Nottingham Forest and now Charlton.
But Brayford, 26, in his third spell under Clough, was quick to salute the Blades boss with whom he worked at previous clubs Burton and Derby.
"Obviously I wasn't here at the start of the season and it was a difficult start," Brayford said.
"So to get to where they are now from where they were in such a short space of time is a tremendous achievement.
"The gaffer's come in not too long ago, but I know he's got us playing the way he wants us to play now.
"I knew his methods from before and the lads have applied themselves to the way he wants them to play.
"When you've got 16 lads all applying themselves in the manner they have done, then you've got a great chance."
The Blades will play Hull over the weekend of April 12 and 13 in their first FA Cup semi-final since losing 1-0 to Arsenal at Old Trafford in 2003. They lost 2-1 to city rivals Sheffield Wednesday at Wembley in 1993.
"It's right at the top of my own personal achievements, I think," added Brayford, who refused to be drawn on his own long-term future.
"You have good feelings throughout your career, but at this moment this is top.
"You see lower-league teams getting the odd good cup result, but we've turned teams over time after time in this run.
"And when we've played the big teams we've always been able to focus on the next game.
"We turned Peterborough over in midweek and it's just great when the momentum is building and you see the fans turn out in their numbers like they have for this.
"It's a sleeping giant, this club, and it's ready to wake up really."