Teenage fly-half Rhys Patchell slotted seven penalties and a drop goal for the home side, who had scrum-half Lloyd Williams sent off after 25 minutes for a tip tackle on opposite number Benoit Paillaugue.
Pierre Berard, Mamuka Gorgodze and Timoci Nagusa all crossed for Montpellier, with the classy Paillaugue kicking 20 points.
The Blues' defeat adds to the gloom that has enveloped Welsh rugby, with the regions managing a collective one win from nine Heineken Cup matches to go with the national side's run of seven straight defeats.
The injury-ravaged Blues, without Sam Warburton, Leigh Halfpenny, Jamie Roberts, Bradley Davies and a host of others, did make a good fist of the opening quarter against a Montpellier side resting several of their first-choice names.
Patchell had the hosts on the board inside four minutes with a penalty and fell just short with a long-distance effort just seconds earlier.
But there were already early warning signs for the Cardiff scrum, an Achilles heel throughout the season.
An emphatic drive from the Montpellier pack allowed Paillaugue to slot a monster 50-metre penalty, and another strong shove gave the scrum-half his second three-pointer after Patchell had briefly put the hosts back ahead.
Patchell, showing a good degree of composure for one so young, replied with another penalty to make it 9-6 at the end of a scrappy opening quarter lacking in creativity.
But the Blues' hopes were dealt a major blow when Wales international Williams was dismissed.
The home crowd howled their derision at referee John Lacey, but the official was left with little choice after Williams lifted opposite number Paillaugue and dropped him on to the back of his neck after the ball had squirted out of a Blues scrum.
Paillaugue dusted himself down to level the scores from the resulting penalty, only for Patchell to again respond, this time with a well-taken drop goal.
But the home side's numerical disadvantage soon told as Montpellier built patiently through the phases to work an overlap on the left for wing Berard to cross, with the reliable boot of Paillaugue adding the conversion from the touchline.
Patchell kept Cardiff in it with yet another penalty, but when Dafydd Hewitt came in at the side of the ruck Paillaugue struck to leave the visitors 19-15 to the good at the break.
The Blues turned round with a strong wind behind them and Patchell used it to bring them within a point with a huge three-pointer from almost his own 10-yard line.
Montpellier should have made the game safe 15 minutes into the second half when Yoan Audrin picked off Patchell's pass.
The young fly-half did brilliantly to haul the winger down, but when the ball was recycled the visitors had a huge overlap, only for Gorgodze's final pass to Berard to be ruled forward.
Paillaugue did at least re-establish the four-point lead with his latest penalty, and the French side's second try followed with 14 minutes to go.
Fly-half Santiago Fernandez gathered in Paillaugue's kick to take Montpellier deep into Cardiff territory. Desperate scrambling defence kept out Berard, but Gorgodze stretched out to dot down through the ruck.
The hosts battled gamely and Patchell quickly slotted another two penalties to get the Blues back to within three points.
But Paillaugue steadied any French nerves before replacement wing Nagusa sealed the win by finishing smartly in the corner.