The home side led 16-13 at the break thanks to an Olly Robinson try and the boot of Jarrod Evans but could never put the visitors away.
Following two tries apiece in the second half, a late John Muldoon score set-up a tense ending, but Gareth Anscombe kept the scoreboard ticking to seal the victory for the home side.
Connacht dominated the opening stages at a bitterly cold Arms Park and set up camp within Cardiff Blues’ territory.
They turned down an early shot at goal to kick to the corner but were unable to win the lineout to rumble over.
However, they remained in the ascendency, despite a thunderous hit in midfield from Willis Halaholo, and opened the scoring on 13 minutes thanks to the boot of Jack Carty.
The penalty sparked the Blues into life and they threatened to strike from the restart after Connacht fumbled in midfield. They were unable to convert the opportunity as the Irish outfit recovered but Jarrod Evans soon levelled the scores.
Cardiff Blues continued to threaten and they claimed the game’s opening try in stunning fashion on 19 minutes.
Evans put Willis Halaholo, who tormented Connacht in September, on the outside of the defence. He released Macauley Cook and although he was hauled down he managed to release Owen Lane with a fine offload from the floor.
Lane, making his PRO14 debut, hit the ball at full pace before returning the ball with an equally impressive offload to Halaholo, who put Robinson over in the corner.
Evans added the extras and soon slotted a second penalty to give his side a 10-point lead after little more than 10 minutes.
Connacht endeavoured to reply and had promising periods of possession around the half-way line but on each occasion they coughed up the ball in the face of Blues pressure.
However, they replied against the run of play on 29 minutes with Niyi Adeolokun intercepting a Halaholo offload in midfield to canter home.
Carty converted the score to bring his side within three-points and minutes later he slotted a penalty to make it 13-13.
It appeared as though the Blues would hit back with a try as half-time approached with Evans ghosting through in midfield and releasing Robinson.
He was unable to find Tomos Williams in support and was hauled down short and despite an overlap to the left, the Blues failed to put the ball through hands.
They continued to press and were eventually rewarded with a penalty which Evans duly slotted for a 16-13 lead at the interval.
Cardiff Blues were put under immediate pressure early in the second half and Connacht attempted to turn the screw up front.
However, the Blues weathered the storm and forced a penalty at a five-metre scrum to clear their lines.
After coming through the period unscathed the home side went on the counter-attack with a series of offloads.
Halaholo continued to cause chaos with Robinson providing a link but when scrum-half Williams attempted an offload, which was palmed forward by Connacht centre Eoin Griffin, a golden opportunity was lost.
Referee Craig Evans was left with little choice but to sin-bin the centre and minutes later the Blues made their numerical advantage count.
They failed to cross from an attacking lineout and scrum but at the third time of asking Morgan mesmerised the Connacht defence as he eased around the Irish defence to crash over.
Gareth Anscombe added the extras to make it 23-13 and he was soon at the tee to convert a try of his own.
It appeared to be another stunning team effort as the replacement fly-half took the ball to the line and then spread it left.
Tom James raced down the outside and put Anscombe away with an inside ball but Evans went to the TMO and adjudged that Nick Williams had taken a defender out as he ruled out the score.
From the ensuing penalty Connacht kicked to touch and rumbled over for a sucker-punch try.
Carty failed to bisect the uprights to cut the deficit further but the Blues were undeterred and claimed their third try on 61 minutes.
They kicked a penalty on the half-way line to the corner and although the ball was lost, Connacht knocked on.
From the ensuing scrum the Blues attacked from first phase with Anscombe again taking the ball to the line and slipping in on the inside to Lane.
He crashed forward as he juggled the pass and bounced up from the floor to score a crucial try on his PRO14 debut, which Anscombe converted.
Again Connacht rallied and although Wales’ Capital Region initially survived, Connacht’s frustration at conceding a penalty boiled over.
Handbags were exchanged and when number eight Williams reacted with a push he was perhaps harshly sent to the sin-bin.
Connacht showed no mercy as Shane Delahunt rumbled over but Anscombe settled nerves with a long-range penalty after a Quinn Roux high tackle that should have received a yellow card.
However, the Galway-based province took advantage of the extra man minutes later, as John Muldoon ran in unopposed with three minutes remaining to set-up a nervy finish.
Connacht, initially looking to play from deep, were forced to kick downfield from the restart. Anscombe gathered and found touch, but a quick Connacht line-out was knocked on. The resulting scrum led to referee Evans awarding Danny Wilson's side a penalty, allowing Anscombe to seal the six-point victory from the boot.