Connacht scored half a dozen tries in Swansea but still fell to an eighth defeat in 13 URC games this season.
They also conceded six tries and in the end had to settle for two bonus points which might not be enough in their battle for a top half finish.
Connacht, who came from 24 points adrift early in the second-half, pounded the Ospreys line for several minutes with the clock in the red but just could find a seventh try which would win the game.
The damage was done in a 27-spell when Ospreys scored 35 points without reply before Connacht staged a third quarter comeback on an evening when they had opened brightly as they tried to keep their outside hopes alive of a top half finish in the URC.
Connacht raced into a 14-3 lead after 14 minutes with Cathal Forde key in creating the opening try which Fiinn Treacy finished well in the left corner and which Josh Ioane did superbly to convert.
But their tendency to cough up points after they score continued with Dan Edwards being presented with an easy penalty to get Ospreys off the mark.
Shamus Hurley-Langton made it 14-3 when he scored under the posts but an injury to Shayne Bolton forced a backline rejig for Connacht with JJ Hanrahan coming on at out-half, Ioane switching to full-back and Piers O’Conor going to the wing.
Ospreys exploited the situation and ran in four tries without reply before the break with Keelan Giles getting two of them and Kieran Hardy and Daniel Kasende also getting in to lead 31-14 at the break.
They pushed the lead out to 24 points four minutes after the restart when full-back Jack Walsh scored but then Connacht hit back through Dave Heffernan and O’Conor, with Hanrahan converting both tries to cut the gap to 38-28 after 50 minutes.
Connacht conceded a second try to Congo native Kasende seven minutes later but struck back with a fine score from Josh Murphy which Hanrahan converted to reduce the margin.
They got within striking distance of the win when Hurley-Langton scored out wide but their late barrage came up just short.
Scorers: Ospreys: Tries: K Giles (2), D Kasende (2), K Hardy, J Walsh. Cons: D Edwards (5). Pen: Edwards.
Connacht: Tries: S Hurley-Langton (2), F Treacy, D Heffernan, P O’Conor, J Murphy. Cons: JJ Hanrahan (3), J Ioane (2).
Ospreys: J Walsh; D Kasende, E Boshoff, K Williams (P Cokanasiga 52), K Giles; D Edwards (T Davies 80), K Hardy (R Morgan-Williams 51, I Hopkins 70); G Phillips (S Thomas 42), S Parry (c), T Botha (B Warren 46), J Ratti (J Fender 46, Hardy 78), A Beard, M Morse, J Tipuric, M Morris.
Connacht: Connacht: P O’Conor; F Treacy, H Gavin, C Forde, S Bolton (JJ Hanrahan 30); J Ioane, C Blade (M Devine 53); D Buckley (T Lasisi 58), D Heffernan (D Tierney-Martin 58), J Aungier (S Illoe 58); D Murray, J Joyce (c) (O Dowling 46); J Murphy, S Hurley-Langton (D O’Connor 76), P Boyle (S Jansen 46).
Referee: A Barrett-Theron (South Africa).
