Celtic League leaders the Ospreys lost for only the second time this season as Connacht produced a shock at the Gnoll.
Ospreys (6) 7
Pens: M Jones (2)
Drop-goal: M Jones
Connacht (7) 10
Try: M Mostyn
Con: P Warwick
The visitors scored the only try of the game when full-back Matt Mostyn crossed midway through the first half.
Fly-half Paul Warwick scored the Irish side's other points with a conversion and a penalty.
Matthew Jones kicked two penalties and a drop-goal for the Ospreys, but the fly-half missed three relatively easy penalties which cost his side dear.
There was more than a suspicion of a forward pass in Connacht's opening try, but Mostyn took no notice of the home crowd's pleas and broke through the Ospreys defence with little difficulty.
Warwick kicked the conversion to put the visitors 7-0 ahead after a dour 20 minutes of play.
Ospreys fly-half Jones had missed an earlier penalty attempt, but found his range with two simple efforts before the interval to bring the league leaders to within a point.
Connacht had been warned by the referee about their indiscipline, and his patience finally ran out when number eight John O'Sullivan conceded their eighth penalty of the match.
He was sent to the sin-bin for 10 minutes, but the Ospreys failed to capitalise on their numerical advantage.
Things went from bad to worse for the home side 15 minutes into the second half when Jones missed two penalties at goal in quick succession.
Connacht, though, were not so wasteful as Warwick added another three points 10 minutes later.
The Ospreys were playing all the attacking rugby and a period of sustained pressure gave Jones the time and space to land a drop-goal.
But the fly-half's miserable night soon continued as the youngster put a fourth kick at goal horribly wide five minutes from time.
Ospreys: A Durston; S Terblanche, D Bishop, E Seveali'i, A Brew; M Jones, J Spice (capt); P James, H Bennett, A Millward, A Newman, L Bateman, J Bater, R Pugh, A Lloyd.
Replacements: R Hibbard, C Griffiths, L Tait, N Bonner-Evans, A Williams, J Hook, D Tiueti.
Connacht:Mostyn; Robinson, Yapp, Downey, McPhillips; Warwick,
Keane; Hogan, Jackman, Bracken, Short, Farley, Swift, Lacey, J O'Sullivan.
Replacements: Halligan, Knoop, Myburgh, Muldoon, O'Loughlin,
Referee: Graeme Hannah (Scotland)