But it didn't look like they would manage the win at half-time as Eric Elwood's men led 17-6 having scored tries through Tiernan O'Halloran and Mike McCarthy.
However the Galway side managed just three second-half points as Alex Cuthbert and Casey Laulala turned the tables, with Scotland fly-half Dan Parks keeping the scoreboard ticking over from the kicking tee.
The home side got off to a flying start as winger O'Halloran powered over after just two minutes after slick handling in the backs.
Parks and O'Connor exchanged penalties before Parks added another three after 15 minutes.
Connacht extended their lead just before the half hour as they forced a lineout deep in the Blues' 22 and from a driving maul second row McCarthy came up with the try.
Parks missed a second penalty of the half on the stroke of half-time as Connacht looked in complete control.
But the Blues came out a different animal in the second half and were right back in it after just a minute.
Laulala broke two tackles on the left before beautifully offloading for Cuthbert who strolled over in the corner with Parks conversion making it 17-13.
And the Blues took the lead two minutes later as Connacht tried to play too much rugby in their own half and prop Rodney Ah You's wild pass was collected by Josh Navidi.
The home side did manage to recover the ball only for John Muldoon's kick to be charged down and eventually Laulala burst through under the posts to put Cardiff in front.
Parks duly added the simple conversion but after Paul Tito was yellow-carded on the half hour, Miah Nikora drew Connacht level.
However the Blues came back and Parks punished some hands in the ruck with 15 minutes to go, before his drop goal ten minutes from time sealed the win for the Blues who move back into fourth place in the table just four points off the top-of-the-table Ospreys.
|Connacht Rugby Score Card|
|Cardiff Blues Score Card|