Out-half Ian Keatley rescued a bonus point for Connacht with the last kick of the game after Jerry Collins had crowned his Ospreys debut with the only try of the game.
Connacht had been reduced to 13 men by the time Collins got the match-winning try at the end of the third quarter.
Scottish referee Graham Knox was booed by the home crowd as he first dispatched Mike McComish to the bin despite the debutant Number eight clearly coming through 'the gate'.
That allowed Dan Biggar to land his second penalty of the evening to push the Welsh side 9-6 in front and it was a lead they never looked like losing when Connacht scrum-half Frank Murphy was binned after Knox ruled that he tried to kill the ball.
The Ospreys kicked to touch and it was no surprise when thirteen-man Connacht were breached as the Ospreys piled over with Collins being credited with the try.
Biggar added the conversion to make it 16-6, but Keatley gave Connacht hope with a penalty 12 minutes from time.
The game was secured four minutes from time when James Hook announced his arrival with a penalty from 40 metres to put ten points between the teams.
However, Keatley then rescued a point for Connacht against an Ospreys team who had started with 13 internationals, when he kept his nerve to steer the ball between the posts from the left-wing.