Ben Blair kicked all 15 points for the Cardiff Blues
Blues: (9) 15
Pens: Blair (5)
Connacht: (13) 13
Pens: Warwick (2)
Five penalties from the lethal boot of Ben Blair saw Cardiff Blues fight back for a narrow victory at the Arms Park.
Connacht wing Matt Mostyn caught the home side cold with a first-minute try, but injured his knee in the process.
Paul Warwick converted and added two penalties to give the battling visitors a 13-9 lead at the break.
But Blair kept his team in touch and the Blues forwards took a grip of the game, setting the platform for two second-half penalties to seal the win.
We know we're not playing with a lot of confidence, but we'll take the win
Man-of-the-match Ben Blair
A minute's silence was observed before the start in honour of Welsh Rugby Union President Keith Rowlands - an ex-Cardiff player and administrator - who died on Saturday at the age of 70.
With the sombre mood seeming to affect the home side they failed to deal with Connacht's switched kick-off, and quickly recycled ball was sent to Mostyn.
The former Newport wing had the power to blast through the midfield defence for a try under the posts, but suffered an injury that ended his match.
Keith Matthews tackles Blues flanker Rhys Shellard
Man-of-the-match Blair pulled a penalty back from the restart, and the New Zealander traded kicks with Connacht's Australian outside-half Warwick for the rest of the half.
The visitors out-scrapped the Blues for a deserved lead at the break, but Blair's third penalty on 39 minutes meant the gap was just four points, 13-9.
The Kiwi missed his only kick of the night early in the second period, as the home side began to take a grip on territory and possession.
With the pressure increasing on Connacht another penalty was conceded on 56 minutes, Blair making no mistake to close the gap to one.
The Cardiff-side's forwards encamped on the Irish line, and when Connacht came in at the side of a ruck Blair finally gave his team the lead with his fifth penalty.
There were still 15 minutes to go, but Connacht could do nothing to loosen the Welsh side's stranglehold on the match.
Man-of-the-match Ben Blair:
"Connacht dominated the first half physically and were the better side.
"But we turned it around in the second period, kept it in their half and got the points to win.
"We're not terribly happy with the performance and know we're not playing with a lot of confidence.
"But we'll take the win and the league points and hope to turn the performances around in the next few weeks."
Blues: Blair, Luveitasau, J. Robinson, Stcherbina, Czekaj, N. Robinson, Evans, Yapp, Goodfield, Roberts, Davies, Jones, Morgan, Shellard, Rush.
Replacements: James for Luveitasau (67), G. Williams for Goodfield (68), Lewis for Morgan (70). Not Used: Gill, Goode, M-Williams, Macleod.
Connacht: Riordan, Mostyn, Hearty, Matthews, Yapp, Warwick, Keane, Wilkinson, Flavin, Sturgess, Gannon, Farley, O'Connor, Ofisa, J. Muldoon.
Replacements: McPhillips for Mostyn (5), McHugh for Warwick (50), O'Loughlin for Keane (61), Hogan for Wilkinson (64), Fogarty for Flavin (61), Rigney for O'Connor (50). Not Used: Strudwick.
Referee: Neil Paterson (SRU)