The 28-8 success saw Blues remain eighth in the table but reduced the gap to the two sides above them, sixth-placed Edinburgh and seventh-placed Munster, to a single point with two games remaining.
Blues' tries came from Josh Turnbull, Ray Lee-Lo and Gareth Anscombe, with Anscombe kicking three penalties and two conversions. Sarel Pretorius scored a try for Dragons, with Dorian Jones adding a penalty.
Cardiff went ahead after seven minutes with a try from Turnbull after Ellis Jenkins had charged down a clearance kick from Pretorius to give Blues a platform in the Dragons' 22. Anscombe missed the conversion but was soon on target with a simple penalty.
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Dragons should have had their first score when they produced the best movement of the first quarter. Hallam Amos beat Dan Fish in the air to collect an up and under from Jones before sending Pretorius away into the home 22. The visitors looked certain scorers but Amos hung onto the ball with two unmarked men outside him.
After 18 minutes Dragons did get on the scoreboard when Sam Warburton was penalised for a shirt tug on Nic Cudd which gave Jones the chance to kick the resulting penalty.
An excellent 40-metre burst from Pretorius split the Blues' defence but Amos was robbed of possession 10 metres out from the try line and the home side were able to clear the danger.
There was little between the teams in the opening half hour with some clever positional kicking from Anscombe being offset by the greater attacking threat from the Dragons.
Anscombe extended his side's lead with a second penalty before Dragons again threatened when Amos beat three defenders in a superb gliding run but the movement fizzled out deep in the Blues 22.
However after 32 minutes Dragons deservedly picked up their first try when Pretorius finished off a driving line-out but they suffered two blows when first Taulupe Faletau was led off with a facial injury before Pretorius was harshly yellow carded for a deliberate knock-on.
Blues took immediate advantage by scoring their second try when Lee-Lo collected a clever chip through from Anscombe, who converted for an 18-8 half-time lead.
Faletau returned after the interval but Blues had lost scrum-half Lloyd Williams at half-time with an injury.
Pretorius returned from the sin-bin with no further damage to the scoreboard but a third penalty from Anscombe put Blues further in front before Rhys Patchell limped off with a hamstring problem.
Anscombe's penalty and his last-minute try and conversion were the only scores of the second half as spirited Dragons frustrated the home side's quest for a valuable bonus point.
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