Dean Dewdney's two tries saw the Blues keep up the pressure on the Celtic League's leading three as they edged out Connacht at the Sportsground.
Connacht (8) 8
Blues (5) 18
Try: Vaughton (2)
Pens: MacLeod (2)
Dewdney touched down in either half while fly-half Nick MacLeod sealed the game with two second-half penalties.
James Downey scored an early try for Connacht while Paul Warwick's penalty helped the Irish side to an 8-5 interval advantage.
The Blues' success was their first win on Irish soil since last March.
Michael Bradley's Connacht men - gunning for their fifth win in six Galway outings - scored off a neat Paul Warwick chip through after only 65 seconds.
The Australian's kick was collected by onrushing wing Conor McPhillips, whose quick step around Craig Morgan put ex-Leinster starlet Downey galloping over to the right of the posts.
Controversy then reigned as Cardiff flanker Nathan Thomas was allowed to boot Warwick's conversion off its tee.
Scottish referee Malcolm Changleng had adjudged that 24-year-old Warwick had begun his run-up despite still being in a crouched position.
That seemed to disrupt Connacht, whose lineout misfired, as the Blues came more and more into the game.
Macleod watched his fifth-minute penalty slam off an upright, but the Welsh were back level on 14 minutes as skipper Tom Shanklin's midfield break put Dewdney over in the right corner.
The young winger was unfortunate not to nab his second two minutes later when knocking on at the base of a ruck, only metres from the line.
Then Macleod saw another kick screw wide before McPhillips' injury-time hand trip saw Shanklin kept out.
Dewdney's sin-binning for preventing a quick release turned the tie in the home side's favour, and Warwick obliged for an 8-5 break lead with his second attempt in added time.
Both kickers missed a series of penalty shots in the second half.
But 21-year-old MacLeod did send over two on 55 and 73 minutes, sandwiching Dewdney's clever 69th-minute kick-through and collect try under the posts.
Connacht: M Mostyn; C McPhillips, D Yapp, J Downey, T Robinson; P Warwick, C Keane; R Hogan, B Jackman, S Knoop, C Short, A Farley, J Muldoon, M Lacey, J O'Sullivan.
Replacements: D McFarland, J Fogarty, P Myburgh, M Swift, T Tierney, D Slemen, M McHugh.
Blues: C Morgan; J Vaughton, T Shanklin (capt), T Davies, D Dewdney; N MacLeod, R Powell; J Yapp, R Thomas, G Jenkins, D Jones, R Sidoli, N Thomas, R Sowden-Taylor, K Schubert.
Replacements: S James, L Thomas, J Bryant, M Molitika, N Budgett, G Williams, M Jones.
Referee: Malcolm Changleng (SRU).