A clinical first half try from centre Matt Healy gave the hosts the advantage at overcast but dry Galway.
A penalty from Gareth Davies was all Blues had to show for a disappointing opening 40 minutes where handling errors and the whistle of the official undermined the visitors.
An early second half score from centre Robbie Henshaw sent the Irish side further ahead before huge drama saw the game change in a flash.
A penalty try as the visiting forwards took control at the set piece and opportunist score from Lewis Jones in three dramatic minutes swung the game back in favour of Blues.
Dan Fish grabbed a third try for Blues to cap a stunning 15 minute spell and the visitors duly held out in some style to claim yet another league scalp as they chase down seventh spot.
Blues made three changes to the side that defeated Scarlets on Judgement Day II with winger Harry Robinson called up for Chris Czekaj while Lloyd Williams started at scrum half.
The only change to the pack came at Number Eight where Ellis Jenkins started, while fly half Davies played his 50th game for the region as Blues looked to build on a hat-trick of wins.
The boot dominated the opening quarter of the game as both teams looked for territory to attack from and refused to run from deep.
Running rugby was at a real premium, but when Blues did get ball in hand they looked dangerous in the opening exchanges as Wales ace Alex Cuthbert cut Connacht wide open with a break from deep.
He floated a long pass wide left to the supporting Dan Fish and the full back kicked ahead, with the cover defence drifting over, only for his effort to have too much pace for Harry Robinson to touch down.
An early penalty from home full back Darragh Leader came back off the right upright before another electric break from deep saw Blues fire the ball through hands.
Macauley Cook and Robinson on the angled run around made big yards down the middle of the field before Fish again rattled into contact. Connacht, though, held firm under pressure and won the scrum after Blues knocked on.
Dafydd Hewitt and Cuthbert combined down the right flank with some neat handling out of the tackle, but once again the move broke down as the home defence solidly held firm.
A penalty, this time for playing the ball on the floor, handed Connacht the opening points of the evening as Nikora confidently slotted the simple kick from in front of the posts.
The lead gave the home side crucial momentum and they looked to turn the screw, after a penalty at the lineout following a tackle in the air on scrum half Kieran Marmion, as Leader kicked for the corner.
The first drive from the hosts was repelled, but Connacht then broke through the middle as a neat grubber kick in behind was collected by a diving Healy and the conversion from Nikora put the hosts 10-0 ahead just before the half hour.
The Irish side suddenly had real zip about their attacking play as they went wide with pace before a high up and under from Nikora was taken by Robinson and marked under huge pressure.
Poor handling in their own half by Blues gift-wrapped Connacht another attacking scrum six minutes before the break.
A free kick at the set piece allowed the alert John Muldoon to take a quick tap and go and drive for the whitewash. The home support thought he was over under the posts, only for the home skipper to be correctly penalised for a double movement.
Blues looked to mount a late charge before the interval after Danie Poolman’s clearly kick flew out on the full. Hewitt broke the defensive line and a penalty for pulling back by Henshaw saw Gareth Davies kick to the corner.
The visitors flexed their muscles up front as they bulldozed up to the line to win another penalty. And this time Davies opted for the three points as he slotted the kick to cut the lead back to 10-3 at half-time.
Blues made a half-time change, as Scott Andrews replaced Taufa’ao Filise, but another penalty for offside within 60 seconds gave Connacht early field position.
The home side rumbled ominously up to the line with only a superb tackle on powerhouse prop Rodney Ah You stopping the giant from close range as Blues put bodies on the line.
But the relentless pressure finally told as the Irish side pounded at the door and found a way through as centre Henshaw touched down from close range to extend the lead to 15-3.
Bradley Davies and Filo Paulo, with storming powerful runs, looked to spark Blues back into life, although Poolman was only inches from intercepting a pass that would have seen him clear as he flew out of the line.
The visitors twice turned down a kick at the posts after winning a penalty, instead opting to go for the corner and hunt the try.
The second attempt saw Blues held up as they drove over before a series of five-metre scrums saw Blues really apply the pressure and shunt Connacht back before the penalty try was awarded.
More drama saw a second try follow just three minutes later as a huge tackle in midfield by Josh Navidi dislodged ball. The always alert Lewis Jones hacked ahead before brilliantly winning the foot race to the loose ball and collecting to touch down. Davies added the extras for the lead.
The game had turned on its head in stunning fashion and Connacht looked to regroup as they won the aerial battle for territory as Dan Parks produced a sublime kick to the corner under pressure.
Front row stalwarts Matthew Rees and Gethin Jenkins were replaced, but Blues continued to enjoy the ascendancy up front as Aly Muldowney was yellow carded as Connacht dragged another huge driving maul down.
Blues duly shunted the resulting scrum up field and flung the ball wide at pace to Fish to score in the left corner for a 22-15 lead with ten minutes left in Ireland.
Connacht, though, were not done and Parks kicked into the corner to set up another home attack in a frantic finish as they kept the ball close and tried to find a way through.
Another yellow card for Connacht, against Eoin McKeon for a dangerous tackle on Hewitt, did not stop a grandstand finale from the hosts. But when Parks went long with a penalty, as he hunted vital territory, his kick ran dead.
A penalty at the scrum gave Humberstone a long range kick to settle any nerves, but his kick sailed left of the target to increase the tension inside the stadium.
But it was Blues that managed the clock in the closing minutes and one finally drive saw the ball run dead and the visitors secure a fourth win on the trot.
Timeline:
22m Nikora p 3-0
26m Healy t Nikora c 10-0
40m Davies p 10-3
45m Henshaw t 15-3
56m Penalty t Davies c 15-10
59m Jones t Davies c 15-17
69m Fish t 15-22
Cardiff Blues: 15 Dan Fish, 14 Alex Cuthbert, 13 Cory Allen, 12 Dafydd Hewitt, 11 Harry Robinson, 10 Gareth Davies (Humberstone 64), 9 Lloyd Williams (Jones 55); 1 Gethin Jenkins (Davies 67), 2 Matthew Rees (Dacey 67), 3 Taufa’ao Filise (Andrews 40), 4 Bradley Davies, 5 Filo Paulo, 6 Macauley Cook, 7 Josh Navidi, 8 Ellis Jenkins (Hamilton 77)
Replacements: 16 Kristian Dacey, 17 Thomas Davies, 18 Scott Andrews, 19 Chris Dicomids, 20 Luke Hamilton, 21 Lewis Jones, 22 Simon Humberstone, 23 Isaia Tuifua
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