The South Yorkshire club beat the weather to get their game on on a snow-covered pitch at Clifton Lane on Saturday.
However, they were limited to a solitary Caolan Ryan penalty as they suffered a 17th defeat from 18 league outings.
Conversely, Bristol had lost their 100 per cent win record when losing to Jersey Reds in their previous match and, initially, they struggled at Rotherham, too.
Pat Lam’s side were only 6-0 ahead at the break after two Ian Madigan penalties.
But an early second-half try from winger Ryan Edwards eased any lingering nerves and Madigan added two more penalties before Luke Morahan’s 76th-minute try made sure.
Rotherham head to in-form Jersey this Saturday, opponents who moved up to fourth with a 52-19 success at London Scottish, their sixth victory in seven games.
Jersey, who saw former South Africa Under-20s full-back Jason Worrall score a hat-trick, against Scottish, replace Yorkshire Carnegie in fourth spot.
Carnegie’s game against second-placed Ealing Trailfinders yesterday was postponed due to the inclement weather.
That could be a blessing in disguise for the Headingley club, who visit leaders Bristol next on Friday.
The only other game to survive the weather at the weekend – aside from Doncaster Knights’ defeat to Richmond – saw Cornish Pirates make it three Championship wins in a row.
They beat Hartpury with an impressive 45-20 success, a fourth victory over those opponents this term.