Connacht chipped away at the deficit throughout the second half through the boot of fly-half Miah Nikora, but were unable to catch Davies' new-look side - running out in their first competitive match since seeing 12 players leave in the off-season.
With a number of internationals missing and injuries to contend with, it took a while for Davies' Blues to find their feet at the Sportsground.
But after a slow opening 20 minutes, the experienced Sweeney broke the deadlock, hitting two penalties in five minutes before his half-back partner Jones - in only his fourth RaboDirect PRO12 start - broke through to score on the half-hour mark, the teenager pouncing after Josh Navidi had disrupted a home line out.
Sweeney's opposite number Nikora had the chance to respond in the final minute of the first-half, but missed from the tee.
It marked a disappointing start for Connacht - coming off the back of a best-ever eight-place finish last season - who were without ex-Blues fly-half Dan Parks through injury.
His stand-in New Zealand-born Nikora - now eligible for Irish selection after three-years as a Connacht resident - kicked the province's first points, five minutes after the break to keep them in touch.
Davies introduced Benoit Bourrust and Lou Reed off the bench for their debuts as conditions steadily worsened at the Sportsground.
Director of rugby Eric Elwood also turned to his bench for impact - bringing on ex-Ulster and Ireland Wolfhound Willie Faloon for his debut in the back-row, and playmaker Matthew Jarvis in the backs.
The substitutions paid instant dividends with Nikora knocking over a pair of penalties to leave his side trailing Cardiff 13-9 with ten minutes to play.
However the fly-half did spurn a brilliant attacking opportunity when he opted to kick over the top into the grateful hands of Dan Fish after Connacht had engineered a promising position inside the Blues 22.
And the Blues - missing nine players through injury - showed tremendous courage to weather a late storm and hang on for victory.
|Connacht Rugby Score Card|
|Cardiff Blues Score Card|