City's centre-back miscued an attempted clearance into his own net as he moved to intercept Tony McMahon's low cross from the right and Clough's side held out for a 1-0 win.
It always looked likely to take an error to settle a game of few thrills or clear-cut chances, but that did not worry Clough, who was delighted to reward his club's fans with a win.
"I though they were brilliant," he said. "To get that level of support when you haven't won away is fantastic and I'm really pleased for them.
"I didn't think we were positive enough in the first half, although we contained Bristol ok.
"You are usually happy with a clean sheet at half-time away from home, but from the start of the second half we did more going forward.
"My two centre-backs did really well. Unfortunately, we lose one of them, Harry Maguire now, because he picked up his fifth caution of the season.
"We have been in the position we scored from a few times and I have been saying to the players that if they just smash the ball across goal they would get an own goal."
City were booed off by their supporters at the end, having seen their methodical approach play hit a brick wall in the shape of United central defenders Maguire and Neill Collins.
Their best chance fell to City skipper Sam Baldock when he ran on to a 36th-minute pass from Jay Emmanuel-Thomas. With Scott Wagstaff better placed square of him, the striker elected to shoot and was denied by goalkeeper George Long.
United's only first half attempt was a 31st-minute free-kick from McMahon, comfortably saved by City keeper Elliott Parish.
The visitors threatened more after the break without testing Parish, while City had what they thought was a late equaliser ruled out for a foul on Long after Baldock's shot had created a melee in the six-yard box.
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