Connacht engineered a last-gasp penalty for Miah Nikora, but he missed from a difficult position near the touchline.
New Zealander Nikora was Connacht's only scorer on the night, converting five penalties with his left boot, while the Ospreys' teenage out-half Matthew Morgan kicked two penalties and a drop goal on his league debut.
Nikora landed two kicks in the opening seven minutes to set Connacht on their way, his second one a cracking effort from the standside.
Morgan slipped when trying to open the visitors' account from a penalty and the Welsh side had a near miss when Ashley Beck knocked on in sight of the try-line. It came from their best attack of the half, with Richard Fussell flying along his wing.
But Connacht had the better of the first quarter, aided by a powerful scrum which yielded a third successful penalty for Nikora.
Some energetic counter-rucking and an advancing maul preceded Morgan's first points, 21 minutes in.
Connacht responded well, moving the ball at pace, and good hands from Dermot Murphy and captain John Muldoon looked to have sent Fionn Carr over for the game's opening try. However, referee Peter Allan ruled that the winger had been held up by Fussell in the corner.
The Ospreys forwards, led by former All Black Jerry Collins, won back some territory in the closing stages and Morgan converted the first of two penalty attempts to close the gap to 9-6 for half-time.
Connacht had to soak up pressure in the third quarter, full-back Gareth Owen missing a penalty from halfway for the Ospreys. The hosts' Kiwi lock Bernie Upton was then dispatched to the sin-bin for throwing a punch.
A lopsided penalty count was hurting the westerners and a snap drop goal from the 18-year-old Morgan, on the hour mark, brought the Ospreys level.
With experienced lock Michael Swift leading the charge, Connacht built for a final crescendo. Nikora coolly slotted two more penalties, after 71 and 73 minutes, to put Eric Elwood's men within touching distance of a precious four points.
But Nikora's countryman Nutbrown had other ideas, breaking through the cover to touch down and help the Ospreys end their two-match losing run in the league.
|Connacht Rugby Score Card|
|Ospreys Score Card|