Dan Fish raced over in an entertaining clash before Tom Brown put the hosts back in the game by dotting down in the far corner with less than an hour played.
A tense and nervy final 20 minutes was finally settled when a moment of magic from man-of-the-match Allen saw him set up Dafydd Hewitt for the try bonus point and win to wild celebrations from the away support.
The game was played in the close and unfamiliar surroundings at Meggetland - roared on by a partisan home crowd of 3,700 - with the usual venue of Murrayfield out of action as a new pitch is installed.
But it was Blues that settled quicker and a big start saw them play with huge tempo and when the home side failed to roll away the opening points came from deadly accurate fly half Gareth Davies.
A high tackle from the restart saw Edinburgh immediately draw level with a confidently struck kick from Jack Cuthbert.
Blues were playing all the attacking rugby in the opening minutes as Tuifua again smashed into contact before Robin copeland and Kristian Dacey offered themselves as runners around the corner.
A high hanging cross-field kick then nearly landed for Cuthbert, but ran dead as the winger looked to latch onto Davies' intelligent effort.
A penalty for side entry at the driving maul saw the hosts kick to the corner. And when they went in field a powerful angle and run from Matt Scott saw him fend off tackles and race in. Cuthbert added the conversion for a 10-3 lead.
The big tackles were now flying in as Cory Allen was up ended in midfield by two men before Filo Paulo looked to unload a huge hit of his own to rock Edinburgh and win back ball as it was coughed up.
Blues were dealt a blow as Lou Reed was forced off, but it did see the return of Wales in international Bradley Davies after just 13 minutes of the game.
The visitors could have easily scored themselves on 15 minutes when full back Dan Fish cleverly kicked in behind and latched onto the loose ball but knocked on as he looked to pick up and dive over the line under pressure.
Blues did however hit back when Kristian Dacey raced down the middle into space after a pop pass from big prop Filise and when the ball was played wide it was Allen who took advantage of the overlap to score. Davies added the difficult extras to draw the teams level.
A barnstorming run from Tuifua and then Copeland put Blues back on the front foot, but Ice was left needed treatment after a huge collision with Jack Cuthbert as he looked to break from deep.
Blues were slicking together some nice attacking moves as Macauley Cook and Lewis Jones linked up on the runaround until momentum was lost by obstruction as the visitors again made yards into the Edinburgh 22.
Dafydd Hewitt replaced a groggy Tuifua just before the half hour as Blues looked to break form deep with Bradley Davies and Cuthbert before the ball was spilled on the floor as the visotors side looked to keep the ball alive.
The Blues game was being played with real pace and Cuthbert injected more as he cut in off his wing and found Dacey supporting on his shoulder.
Gareth Davies turned down a kick at goal to go for the corner, after a penalty for side entry at the ruck, and the decision was justified as lock Paulo then drove over from a huge lineout drive. Gareth Davies again added the difficult extras and Blues led 17-10 at the break.
Both teams looked to start half on the front foot with Allen and Bradley Davies combining to shunt Jack Cuthbert into touch as he looked to dance into space from deep.
But it was Blues who went further ahead when slick hands down the left freed Fish and he raced in down the touchline to score a third away try.
Fish thought he was away again three minutes later after Blues stripped the ball in the tackle, but back came play for a marginal forward pass in the tackle from Alex Cuthbert.
Edinburgh needed a spark and got it when Denton twice barged into tackles and the home side finally found a way over in the corner with the try going to Tom Brown after the referee consulted with his touch judge. Jack Cuthbert's brilliant conversion cut the lead to seven.
Blues looked to play territory as Fish kicked smartly into the corner with 15 minutes remaining as the tension started to grow at Meggetland.
The visitors secured a penalty with a huge drive up field from the lineout, but then went wide right and almost broke through via Cuthbert and Allen before the ball was dropped in the tackle.
The game entered the final ten minutes still on a knife edge and with Edinburgh looking to run the ball from everywhere and turn the screw. Blues in response saw Cuthbert cut a great line but he was taken down as he entered the 22.
A moment of magic settled the game as Blues showed great composure before Allen produced a delightful pass out of the tackle to deceive two men and allow Hewitt to dot down in the corner for the try bonus.
A late comeback from Edinburgh saw Bradley Davies yellow carded and the home side score again through Cornell Du Preez but Cuthbert saw his kick hit the crossbar.
03m G Davies p 0-3
05m J Cuthbert p 3-3
11m Scott t J Cuthbert c 10-3
19m Allen t G Davies c 10-10
34m Paulo t G Davies c 10-17
42m Fish t G Davies c 10-24
48m Brown t J Cuthbert c 17-24
75m Hewitt t 17-29
79m Du Preez t 22-29
Cardiff Blues: 15 Dan Fish, 14 Alex Cuthbert, 13 Cory Allen, 12 Isaia Tuifua (Hewitt 28), 11 Chris Czekaj, 10 Gareth Davies, 9 Lewis Jones (Williams 70); 1 Gethin Jenkins, 2 Kristian Dacey (Rees 52), 3 Taufa'ao Filise (Andrews 52), 4 Lou Reed (Davies 13), 5 Filo Paulo, 6 Macauley Cook, 7 Josh Navidi (Jenkins 75), 8 Robin Copeland
Replacements: 16 Matthew Rees, 17 Thomas Davies, 18 Scott Andrews, 19 Bradley Davies, 20 Ellis Jenkins, 21 Lloyd Williams, 22 Simon Humberstone, 23 Dafydd Hewitt