A last-minute try from scrum-half Dean Dewdney secured a bonus-point win for the Blues at Cardiff Arms Park.
Blues (11) 31
Tries: Shanklin, R Williams, Morgan, Dewdney
Pens: Macleod (3)
Connacht (3) 6
Pens: Warwick (2)
Tom Shanklin's first-half try gave the Blues the advantage, two Paul Warwick penalties keeping Connacht alive.
But the Blues exploded in the last 20 minutes, a Martyn Williams break freeing Rhys Williams for the next try.
Weak tackling from Ted Robinson let Craig Morgan in for a score on the left, before the Rhys Williams-Martyn Williams combination sent Dewdney in.
The visitors had their back-line disrupted by an early knee injury to Matt Mostyn, and the Blues took advantage as a fine
three-quarter move involving Shanklin, Rhys Williams and Morgan freed Nick Walne.
He was hauled down five metres short, with Morgan on his shoulder waiting for the try-scoring pass.
Nick Macleod opened the scoring with a 14th-minute penalty as the home side's pressure mounted.
Ten minutes later the first try came, clean ruck ball in the shadow of the Connacht posts being spun to Macleod, who freed
Shanklin in the corner with a beautifully weighted pass.
Macleod and Warwick exchanged penalties to give the home side an eight-point advantage at the end of a lacklustre first
half dominated by the whistle of referee Malcolm Changleng.
Connacht tried to get back into the game through their rolling maul, with No 8 John O'Sullivan to the fore of a rugged pack
that repeatedly stole the home side's attacking ball.
Macleod and Warwick exchanged penalties, meaning the game was still up for grabs going into the final quarter.
But the result was sealed when lax defence at a maul allowed Martyn Williams to break clear into the Connacht 22.
He fed the supporting Rhys Williams, who sprinted clear under the posts for a converted try.
Morgan used his pace and power for the third try
Blues lock Robert Sidoli was sin-binned for going over the top of a ruck, having earlier been warned for killing the ball.
But Connacht failed to take advantage, and Robinson's mistake allowed Morgan in on the left wing for the Blues'
They had five minutes left to chase the bonus-point try, and as their furious attacks mounted it was Rhys Williams who had
the gas to break the line.
He found Martyn Williams in support, who fed Dewdney for a clear five-metre sprint to the line.
"Our internationals led the way, Rhys Williams was outstanding and that was a Welsh captain's performance from Martyn," said Blues coach Dai Young.
"That was the best we've played as a team, our approach work was excellent on occasions although some of our finishing let us down.
"I was mystified by Sidoli's yellow card when Connacht had spent most of the game slowing our ball down.
"It was hard to build tempo, but they cracked towards the end."
Blues: R Williams; N Walne, T Shanklin, T Davies, C Morgan; N Macleod, D Dewdney; G Jenkins, G Williams, M Jones, D Jones, R Sidoli, N Thomas, M Williams (capt), K Schubert.
Replacements: K Fourie, A Lewis, H Senekal, R Sowden-Taylor, R Smith, L Thomas, J Vaughton.
Connacht: D Slemen; N O'Brien, M Mostyn, M McHugh, C McPhillips; P Warwick, T Tierney; D McFarland, B Jackman, P Bracken, P Myburgh, A Farley, P Neville, M Lacey, J O'Sullivan.
Replacements: J Fogarty, A Clarke, M Carroll, M Swift, E Elwood, C O'Loughlin, T Robinson.
Referee: Malcolm Changleng (Scotland)