Blues made a dream start to the last eight tie as top try scorer Lloyd Williams again showed his eye for the line with a try inside the opening five minutes of a pulsating derby encounter.
Fly half Gareth Anscombe also crossed the whitewash before home wing Hallam Amos darted over to cut the lead to 14-10 at half-time.
A powerful second half surge from the hosts saw replacement back row Nic Cudd put Dragons ahead before a penalty try sent them clear.
Josh Navidi’s late try gave Blues hope but despite a late rally it was the Dragons who moved into the final four and left the visitors ruing what might have been after such a strong start.
Blues were forced into a late change in Newport as Tavis Knoyle moved into the inside centre jersey, with Joaquin Tuculet ruled out with a hand injury following a fitness test on the morning of the game.
A cauldron of noise and fireworks met the two team as they ran out for the big European derby. And it was Blues that made an explosive start as Josh Navidi and Lou Reed rattled into contact while Gavin Evans charged down a clearing kick from Jason Tovey as Dragons conceded a five-metre scrum.
Back row dynamo Navidi made a dart for the line and Alex Cuthbert was just held up before scrum half Williams threw a dummy and squirmed over for the opening try.
The home side looked to respond and a well-drilled Blues defence held firm, with the new midfield of Evans and Knoyle combining, until a penalty at the breakdown gave Tom Prydie a simple kick to put Dragons on the board.
Taulupe Faletau was an early replacement for Dragons, with Andrew Coombs forced off after 12 minutes.
Amos looked to spark an attack from deep, but his kick ahead ran dead and allowed a composed Dan Fish to mop up danger. And solid work on the floor won a penalty against Dragons just as they started to dominate territory.
A hectic passage of play saw Blues almost break through courtesy of Knoyle’s direct running and neat offload to Fish before Amos almost picked off an intercept pass but had his trailing leg tackled by Evans as he looked to breakaway.
Sublime hands from Scott Andrews fed Cuthbert to race away down the right wing, but his delayed inside pass failed to find the hands of support runner and early try scorer Williams.
More pressure saw Fish and Evans this time combine down the left, Knoyle add direction in midfield before Williams, Warburton and Cuthbert, who was held up, all made a bash for the try line from close range.
A solid scrum then fed Anscombe who dropped his shoulder, cut inside the last man and had the power to touch down under huge pressure from retreating Dragons for a 14-3 lead.
Blues were given a let-off from restart as the kick off went long before the visitors lost a lineout just metres from their own line.
Solid defensive work managed to repel the danger, but moments later they were breached as a cutting backs move saw Amos race over. Referee JP Doyle went to the TMO to check for obstruction but allowed the try.
Wing Prydie then had another shot at the posts just before the break, but his with a penalty from halfway drifted wide left of the target as Blues took a lead into the break.
A half-time change for Blues saw Sam Hobbs replace Gethin Jenkins up front while Dragons lost injured back row James Thomas moments after the restart replaced by Nic Cudd.
The home side turned down a penalty as Jason Tovey kicked to the corner and the decision paid off as replacement Cudd immediately got in on the action as he stretched over to score as Dragons took the lead for the first time by a point.
Suddenly it was all Dragons as another kick to the corner saw them drive with menace but lose the ball as it wheeled around.
A charged down kick on Williams close to his line saw him pull his opposite number back as he looked to dot down and referee Doyle immediately yellow carded the visiting scrum half and awarded a penalty try to a thunderous home reception.
Nineteen unanswered points had Blues on the rack and the coaching team threw on Kristian Dacey and Macauley Cook to try and spark a revival.
A powerful lineout drive secured a penalty, but Anscombe failed to find touch and back came the hosts as another charged down kick gave them a penalty that Prydie should have slotted yet pushed wide.
Another penalty for a deliberate knock-on gave Dorian Jones the chance to take over kicking duties and he extended the lead to 25-14 into the final quarter.
It was now or never for Blues and they forced a penalty after the forwards drove up field and Anscombe kicked to the corner.
A crooked throw saw the first chance lost as time started to ebb away. But a series of scrum penalties saw Blues finally apply pressure and force their way over with Navidi touching down to cut the lead to 25-21 with five minutes left.
A late fight-back saw Blues again claim back possession and drive up field to force some anxious moments for the home fans. But with number out wide, Blues kept the ball tight and one last penalty from referee Doyle this time went against them and sparked home celebrations.
05m Williams t Anscombe c 0-7
09m Prydie p 3-7
26m Anscombe t Anscombe c 3-14
36m Amos t Prydie c 10-14
46m Cudd t 15-14
52m Penalty t Prydie c 22-14
68m Jones p 25-14
75m Navidi t Anscombe c 25-21
Cardiff Blues: 15 Dan Fish, 14 Alex Cuthbert, 13 Gavin Evans, 12 Tavis Knoyle, 11 Richard Smith, 10 Gareth Anscombe, 9 Lloyd Williams; 1 Gethin Jenkins (Hobbs 40), 2 Matthew Rees (Dacey 53), 3 Scott Andrews (Jones 58), 4 Jarrad Hoeata (Cook 53), 5 Lou Reed, 6 Josh Turnbull (Jenkins 72), 7 Sam Warburton, 8 Josh Navidi
Replacements: 16 Kristian Dacey, 17 Sam Hobbs, 18 Adam Jones, 19 Macauley Cook, 20 Ellis Jenkins,, 21 Lewis Jones, 22 Simon Humberstone, 23 Gareth Davies