Matt Mostyn collected the only try of the game - mid way through the first half as the men from the West of Ireland moved themselves into sixth place.
However the Ospreys had every chance to have taken the points with Matthew Jones missing five penalty kicks and also spilled the ball right on the Connacht line.
Mostyn's trey was converted by Paul Warwick and although Jones had cut the lead to a single point by the break through two penalties the Ospreys had struggled up front - particularly at the lineout where Huw Bennett was having difficulty hitting his jumpers.
After the break Warwick slotted a penalty, only for Jones to reply with a well-taken drop-goal. But the Ospreys outside-half missed a chance to win the game when he pulled wide a routine shot 25 metres out with six minutes left on the clock.