Striker Shaun Miller headed home in the last minute of extra-time to sink Rene Meulensteen's Barclay's Premier League strugglers and Clough said the Blades supporters at Craven Cottage had lifted his players.
"What a night for our 2,000-plus fans, who made the long trip south on a dreadful evening," said Clough.
"I don't think they stopped singing all night and again in the dressing room afterwards the players were all saying how that backing gave them that extra ounce of energy when a number of lads were clearly running on empty.
"It is a cliche, but the continued support on nights like this makes the difference and I thought it was a terrific gesture by the club to put on 23 free coaches as a reward for their support throughout the season to date."
Clough's side, who beat Aston Villa in the third round, will play either Nottingham Forest or Preston for a place in the quarter-finals.
- Sports & Recreation
- Rene Meulensteen