Four crisply-struck penalties from Ian Keatley, who was playing his last home game for the province, gave Eric Elwood's men the edge.
But the improving Blues enjoyed most of the territory in the second half and with Ray Ofisa and Gavin Duffy earning yellow cards, Connacht could not hold out.
Dan Parks landed three penalties after the break and Wales lock Davies stormed over for the clinching try, six minutes from time.
Keatley's first points came in the third minute, with his kick just nipping inside the left hand post. Cardiff, who brought Ma'ama Molitika into the back row as a late change, then got their big ball carriers involved.
Ofisa put in a try-saving tackle on Gareth Williams and Connacht stand-in captain Johnny O'Connor was ruled offside at a ruck, before Parks kicked Cardiff level.
The Blues gave away a succession of penalties at the breakdown and after Davies was pinged, Keatley sent over a fine kick from just inside the visitors' half.
All the early pressure came from Connacht with Sean Cronin, Niva Ta'auso and the province's new record try scorer Fionn Carr probing around the fringes. However, David Young's side soaked it up and threatened on the counter.
A penalty was called against hooker Williams for backchat and Keatley duly converted the right-sided kick for a 9-3 lead, with a quarter of an hour remaining in the half.
Cardiff hit back with a well-taken try from Laulala. The former All Black took a great angle onto a pass in the Connacht 22, glided past Bernie Upton and Mike McCarthy and his clever footwork created enough space for him to stretch over and score by the posts.
Television match official Dermot Moloney confirmed the grounding and Parks added the extras, but some cohesive play from his pack allowed Keatley restore Connacht's lead late on.
Molitika was held up over the try-line as the Blues made a strong start to the second period, Parks mopping up with a neat penalty from the right.
Ofisa was unfortunate to be sin-binned after he upended Jamie Roberts in a tackle that was not malicious. Parks did the necessary with the resulting kick and added another to move the Welshmen into a 19-12 lead.
Cardiff upped their physicality, squeezing Connacht in the scrum and there were some huge tackles put in. Taufa'ao Filise had a bone-crunching hit on Rodney Ah You and McCarthy grounded Molitika with a powerful challenge.
Parks missed two penalties either side of Duffy's sin-binning for a ruck offence in his 22. And the killer blow came when Davies escaped McCarthy's grasp and handed off Ta'auso on the way to the try-line.
Connacht attacked for the remainder and both Ta'auso and John Muldoon, who came on as a half-time replacement, went close to grabbing a try from close range. But they missed out on a losing bonus point and have slipped to ninth in the table..
The result keeps the Blues in contention for a semi-final place. They are now just two points behind the third-placed Ospreys ahead of their clash with Benetton Treviso next Thursday.
|Connacht Rugby Score Card|
|Cardiff Blues Score Card|