Connacht travelled on the back of recent home wins over the Scarlets and Glasgow Warriors, and were quietly confident of troubling the defending champions.
They did just that, leading twice in the first half and almost earning a late losing bonus point.
In the end, the visitors went unrewarded as the Ospreys showed a clinical touch in running in tries through backs Richard Fussell, Tom Prydie, Sonny Parker and replacement hooker Mefin Davies.
As they push to make the Magners League play-offs, Davies garnered his side's fifth try-scoring bonus point of the campaign in the dying minutes. The veteran front rower powered over on the end of a lineout drive.
Connacht had a try in each half from experienced forwards Johnny O'Connor and Michael Swift, with out-half Ian Keatley and his replacement Miah Nikora kicking the rest of the points.
Keatley gave the visitors an early advantage, punishing Ian Evans for straying offside, and Connacht held onto their 3-0 lead until Dan Biggar, the league's top points scorer this season, sent over a 19th minute drop goal.
The Ospreys enjoyed most of the territory in the opening quarter, but Biggar missed a left-sided penalty and Connacht's defence was well organised. Jerry Collins, Evans and Hanno Dirksen all took the ball in the lead-up to Biggar's drop goal, which he slotted while the hosts had a penalty advantage.
Connacht responded almost immediately, using a solid lineout as an attacking platform. After Keatley initiated an attack, winger Brian Tuohy made good yardage before passing for the supporting O'Connor, the province's stand-in captain, to make it over in the right corner for an unconverted try.
The Ospreys failed to convert a couple of try-scoring chances, both Dirksen and centre Ashley Beck went close. Connacht infringed in the scrum, with Biggar kicking the resulting penalty for an 8-6 deficit.
In the 33rd minute, the Welshmen broke into the lead for the first time. Scrum half Rhys Webb set the outside backs in motion in a free-flowing attack, and full-back Fussell hit the line at pace to slice the Connacht defence open and race in under the posts.
Biggar converted and added a penalty with the last kick of the first half, moving the home side into a 16-8 lead at the break.
Connacht had a decent spell of possession when play resumed in the second half, yet Eric Elwood's men were thwarted by the Ospreys' committed defending and they were quick to punish their opponents on the counter attack.
The Ospreys again moved the ball at pace and a neat line of passes saw them finish off an overlap on the left wing, with Parker passing for Prydie to score his first Magners League try. It was a belated birthday present for the Wales-capped teenager, who only turned 19 last Wednesday.
Biggar's missed conversion was followed by a penalty miss from Keatley, and the Ospreys scrambled well to deny Connacht after strong running from two of the province's Ireland internationals, O'Connor and Gavin Duffy.
Keatley was back on target in the 55th minute to punish the Ospreys for a scrum infringement and reduce the arrears to 21-11. Just four minutes later, Parker was up in support to score his side's third try after replacement prop Ryan Bevington had been held up just short.
The wind was proving difficult to read for the place-kickers and Biggar (conversion) and Keatley (penalty) both missed further kicks. Connacht showed great resilience as their forwards took the game to the Ospeys in the final quarter.
Number 8 Ray Ofisa knocked on with the try-line in his sights, but the westerners did not lose heart.
They had the Ospreys on the bink of leaking a penalty try - referee Stefano Penne warmed them after two five-metre scrums were reset - but the home pack survived momentarily.
Ofisa picked off the base of the scrum and was tackled short of the whitewash, only for lock Swift to pick up and score from a couple of metres out. Nikora's successful conversion got Connacht back within eight points at 26-18.
Connacht continued to look dangerous with ball in hand and it took a well-timed challenge from Fussell to prevent a try for Troy Nathan. Nikora missed a hotly-disputed penalty as Connacht's chances of taking a losing bonus point slipped away.
The Ospreys mustered a final surge through a series of lineouts. With Swift dispatched to the sin-bin for coming in at the side of a ruck, they took full advantage to send Davies crashing over for his first try for the region. Biggar added the extras for a 33-18 final scoreline.
Speaking after the game, a disappointed Eric Elwood said: "To leak four tries in the manner we did was disappointing. We knew we had an opportunity today, we knew we had to meet the physical challenge of the Ospreys and we didn't bring that to our game.
"We had an opportunity of a bonus point at the end, missed our kick and then they got a bonus try at the end.
"The Ospreys are good at what they do, they took their chances. We've a big game against Benetton Treviso next Friday so we'll look ahead to that and won't hang around feeling sorry for ourselves."
|Ospreys Score Card|
|Connacht Rugby Score Card|