Tries early in the second half by full-back Matthew Morgan and wing Alex Cuthbert put the Blues in control after a tight opening 40 minutes.
A much-changed Bath side put up a fight in the first half but had only Tom Homer's penalty to show for it against three by Steven Shingler.
Scott Andrews' late opportunist try emphasised Blues' superiority.
The win saw Blues open a five point lead at the top of Pool Four and they face Bath again on Thursday, 15 December at the Recreation Ground.
The Welsh region dominated the early stages with Shingler nailing two early kicks before Homer hit-back for Bath.
The under-strength English visitors had their chances, with England wing Semesa Rokoduguni going close.
Former Scarlets fly-half Shingler kicked a third penalty but another long-range effort fell short and Bath survived a Blues breakout led by Rey Lee-Lo and Alex Cuthbert which saw Josh Navidi hauled down just short of the try line on the stroke of half-time.
Blues head coach Danny Wilson said: "We've come of age a little bit in terms of maturity and game plan today.
"We've played a kicking game and fed off their loose kicks with Matthew Morgan, so it's a pleasing pleasing result in very difficult conditions at a time when we needed a win.
"We've taken a lot of criticism that we haven't defended well but that was a good shift today to hold that team to three points."
Cardiff Blues: Matthew Morgan; Alex Cuthbert, Rey Lee-Lo, Willis Halaholo, Blaine Scully; Steven Shingler, Lloyd Williams; Rhys Gill, Kristian Dacey, Taufa'ao Filise, George Earle, James Down, Seb Davies, Sam Warburton (capt), Josh Navidi.
Replacements: Matthew Rees, Brad Thyer, Scott Andrews, Jarrad Hoeata, Kirby Myhill, Tomos Williams, Nicky Robinson, Cory Allen.
Bath: Tom Homer; Semesa Rokoduguni, Max Clark, Ben Tapuai, Jack Wilson; Dan Bowden, Darren Allinson, Beno Obano, Ross Batty, Max Lahiff, Charlie Ewels, Elliott Stooke, Matt Garvey (capt), Tom Ellis, Paul Grant.
Replacements: Jack Walker, Nick Auterac, Kane Palma-Newport, Guy Mercer, Michael van Vuuren, Will Homer, Adam Hastings, Jeff Williams.
Referee: George Clancy (IRFU)
Cardiff Blues maintained their 100% record in the European Challenge Cup as they inflicted a first defeat on Bath.