Scottish winger Ryan Flynn and on-loan Derby defender John Brayford struck second-half goals in the space of two minutes to seal the Blades a place in the last four for the first time since 2003.
Clough arrived at Bramall Lane in October not long after being sacked by Derby with the club lying second from bottom in Sky Bet League One.
A 2-0 win against Peterborough on Tuesday was their eighth straight win in all competitions - they are now 11th - and a ninth consecutive victory has ensured they become the first club from the third tier to reach the semi-finals of the competition since Wycombe in 2001.
"There's not an awful lot of satisfaction for me personally," said Clough, a losing finalist at Wembley in the 1991 FA Cup final as a player with Nottingham Forest.
"I don't have a chart at home where you rank the games. I don't honestly, it's not about that.
"It's about the players and supporters, especially this, celebrating together."
The Blades have beaten Aston Villa, Fulham and Nottingham Forest in previous rounds and the momentum was clearly with them in front of a sell-out Bramall Lane crowd of 30,040.
Charlton had scuppered the chance of a Steel City quarter-final by beating Sheffield Wednesday in the previous round and gave as good as they got before the game's turning point in the second half when winger Callum Harriott spurned a golden chance to put the Londoners ahead, dragging his 10-yard shot wide from in front of goal.
"We were nervous in the first 20 minutes and it was important to get in 0-0 at half-time," Clough added.
"The atmosphere, it saps you when you're not used to it and a lot of players playing in this sort of thing for the first time, this atmosphere takes your energy and it takes your breath away at times.
"To get to this stage as a League One club in the semi-final is an incredible achievement.
"No easy draws as well along the way, going down to Fulham and Villa Park and we've had a few breaks like here when the lad missed that chance.
"It's essential that you get that little break at certain times."
When asked what was behind his side's impressive turnaround, the former Burton boss replied: "Hard work. The work-rate of the players.
"We had a sit down after the Crewe game, which was February 1. We lost 3-0 and were awful that day and we had a bit of a clear-the-air, went down to Fulham and won and we've gone on from that point."
The Blades' last trip to Wembley in the FA Cup was in 1993 when they lost out to city rivals Wednesday in the semi-finals.
"We're going to try (and win it)," added Clough.
"You look at the teams left in and it's a little bit scary, who ever you get, but it's about the financial boost and everybody enjoying Wembley.
"But we're not going there for a day out. You go to the seaside for a day out, not to Wembley."