The reigning champions led by just a John Cooney penalty after the scrum-half and Tom James exchanged tries in a frantic first-half.
But the Blues were unable to build upon the score following the restart as the Irish province turned the screw.
Cooney kept the scoreboard ticking and Bundee Aki finished a try against the run of play to secure the win.
Danny Wilson’s men refused to throw in the towel and fought to the death in bitterly cold conditions but they were unable to finish the chances they created.
While both teams demonstrated their attacking intent early on, Connacht dominated territory and set up camp in the Cardiff Blues half.
They enjoyed numerous sustained attacks, and retained possession superbly but the Blues defence held firm. But while Wilson’s team were defending with venom they were unable to work their way out of their own half and the pressure eventually told.
Following a Connacht counter-attack which saw a pair of offloads release Ily Stacey, who kicked through, Tomos Williams raced back to concede a scrum five and from the ensuing set-piece John Cooney touched down.
He converted the try to give his side a 7-0 lead but the Blues responded perfectly from the restart with James picking off Carty’s floated pass for his seventh try of the season.
Shingler knocked over the extras to draw level and his side were sparked into life. A rampaging charge from Willis Halaholo laid the foundations for a sustained attack but the Blues were unable to turn pressure into points.
Shingler was off-target on 25 minutes after the pack flexed their forward muscle and back stormed Connacht.
They almost broke through but Cardiff Blues scrambled well and forced a turnover penalty to clear their lines.
The game continued at a frenetic pace with the Blues creating plenty of half chances but unable to hold onto possession for sustained periods.
And Connacht finished the half in the ascendant with Cooney slotting a late scrum penalty to bring the scores to 10-7 at the interval.
The Irish province controlled the opening period once again and Cooney slotted a second penalty on 49 minutes after the Blues were penalised for obstruction.
Connacht took a major stride towards victory on 57 minutes when Bundee Aki crashed over the whitewash following a turnover.
Niyi Adeolokun hacked through on the counter-attack and Williams was unable to collect. The wing regathered the ball and although he was brought down possession was maintained and with a huge overlap Aki touched down.
The hosts failed to add the extras and Cardiff Blues desperately attempted to back but they were unable to breakthrough as the clock ticked down.
Eventually it looked as though they had manufactured a score thanks to a fine Alex Cuthbert break with his first touch.
The wing sliced through the Connacht defence and looked to find Rey Lee-Lo. The Samoan centre went tumbling to the ground and the ball instead found its way to Matthew Morgan.
He beat one defender and crossed the whitewash but TMO relays showed the ball was slapped out of his hand and Owens did not deem it fit to award a penalty after Lee-Lo was taken out.
The Blues dominated the closing stages but were dogged by errors and turnovers at key moments in difficult conditions.