First-half tries from Ellis Jenkins and Blaine Scully helped Wales’ Capital Region into a 14-3 lead at the interval and Jarrod Evans kept the scoreboard ticking after the break.
The victory avenged an opening day defeat to the Scots and stretched the Blues winning streak in first team competitions to an impressive eight.
Danny Wilson’s side will now face Pau, who beat Stade Francais last night, in the last four with a Cardiff Arms Park semi-final set for the weekend of April 20.
Cardiff Blues made a promising start to the encounter with a clever Tomos Williams kick putting pressure on the hosts.
Rey Lee-Lo was also causing problems with the ball in hand and it his side that had the first opportunity to open the scoring when Grant Gilchrist failed to roll away.
Evans’ penalty was on target but fell agonisingly short of the cross-bar. The Blues continued to enjoy the majority of territory but it was Edinburgh that were benefitting from the early refereeing decisions.
The Scots broke the deadlock on 18 minutes thanks to the boot of Jaco van der Walt after Josh Navidi was perhaps harshly penalised at the breakdown.
But it did not take long for the Blues to respond after Williams and Gareth Anscombe combined. The Wales international almost broke clear after he hit the outside but he was scragged by the Scottish cover.
The Blues maintained possession and Lee-Lo again made yards, this time down the left. Two phases later Evans dinked a cross-kick to the corner, which bamboozled Nathan Fowles, who bizarrely headed the ball and Jenkins pounced.
Evans converted from the touchline to make it 7-3 after 23 minutes and Wales’ Capital Region soon claimed a second try.
The try once again came from a kick, this time a probing grubber from Anscombe, which Blair Kinghorn failed to deal with.
Lee-Lo snaffled possession and offloaded to Scully, who cut inside to split two defenders, leaving the travelling support in raptures with chants of USA! USA! USA! reverberating around BT Murrayfield.
Evans once again slotted the extras to give his side a 14-3 advantage and it could have been more as they continued to cause Edinburgh problems.
Turning the Scottish defence was clearly a pre-planned tactic with Williams, Evans and Anscombe all threading intelligent kicks through expertly.
And it was an Evans chip that unlocked the Edinburgh defence once more with Anscombe doing remarkably well, beating two defenders to collect.
Possession was moved to scrum-half Williams on the right and it looked as though he would wriggle clear but he lost possession in the process.
Alarm bells rang on several occasions in the closing stages of the half as mistakes crept in but the Blues hung onto their 11-point lead going into the break.
Evans had the opportunity to nudge the Blues further in front shortly after the second-half got underway but his long-range effort fell short and minutes later Van der Walt cut the lead with a shot of his own.
However, almost immediately after the restart, the Scots were penalised for going off their feet at the breakdown and Evans made amends to restore his side’s 11-point advantage.
Cardiff Blues’ defence was also pressuring the home side into mistakes and a spot rush from Halaholo saw the ball come loose. Owen Lane hacked on but he was unable to win the race to the ball and Edinburgh managed to exit.
Evans kept the scoreboard ticking with his second penalty to open up a 14-point advantage but there was still a tense finale.
Edinburgh managed to establish a stranglehold in the final quarter and matters were made worse when try-scorer Jenkins was harshly penalised.
The Scots laid siege for the best part of 15 minutes but the tenacious Cardiff Blues held them at bay.
Edinburgh rallied once more in the final five minutes, but Danny Wilson's side once again displayed a heroic effort in defence to secure the victory and a home semi final.