Man-of-the-match Shane Williams scampered through for a ninth minute opener, converted by assured fly-half Dan Biggar.
Glasgow Warriors got off the mark, two minutes into the second half, however Biggar sandwiched a classy try from James Hook with two more penalties as the hosts ran out 15-point winners.
Glasgow put up a brave fight, but with league top scorer Dan Parks missing three place-kicks - one from inside his own half - and attack after attack thwarted by the Ospreys' rock solid defennce, an opportunist try from hooker Fergus Thomson proved to be their only score.
The battle of the back rows had dominated the build-up to this semi-final clash, with double Grand Slam winner Ryan Jones and All Blacks' Jerry Collins and Marty Holah facing the collective might of the 'Killer Bs' - Kelly Brown, John Barclay and Johnnie Beattie.
The Scotland-capped trio had their moments, but the Ospreys back row unit had the bigger say with Holah swarming all over the breakdown and a bruised and battered Collins putting in some huge hits.
The Killer Bs could not exert their usual influence as the Warriors were pressed into mistakes and the Ospreys, with Biggar kicking well and his forwards winning the collisions, were able to take the initiative.
The Ospreys were unchanged from last weekend's bonus point win over the Newport Gwent Dragons, while Glasgow made five changes to the side that gained a five-point return in Llanelli.
Scotland captain Chris Cusiter was sidelined by a knee injury, with Mark McMillan taking his place at scrum half, and Max Evans, Ed Kalman, Tim Barker and John Barclay also returned to the starting line-up.
Lineen's charges travelled south chasing their fifth successive league win as the best away side in this season's competition.
The kick-off was put back by ten minutes due to traffic problems which delayed the Glasgow bus and four of the Ospreys players.
The shorter warm-up did not seem to hamper the Ospreys and Biggar, in particular, who kicked two well-weighted touchfinders to get his side on the front foot.
Off the second of those, Alun Wyn Jones, who was making his 100th Ospreys appearance, stole a lineout but Biggar pushed a subsequent drop goal attempt to the right and wide.
They made the breakthrough, though, just three minutes later. It was a try out of nothing which caught Glasgow cold. Glasgow lost the ball in contact, the loose ball was tapped up by Hook for Williams to gather and break over the halfway line, slipping out of a tackle from McMillan as he danced along the touchline and ran over unopposed to the left of the posts.
There was a suspicion that the Welsh wing wizard had put a foot in touch, but television replays gave him the benefit of the doubt and Biggar converted to make it 7-0.
Parks' first penalty attempt, awarded after the Ospreys had dropped a scrum, fell just short of the posts and the tension that comes with such a high stakes game bubbled over when Holah and Glasgow winger Rob Dewey clashed off-the-ball.
The Ospreys tested the Glasgow defence once more when they won turnover ball at a lineout and Huw Bennett and Mike Phillips threatened in quick succession. Glasgow's solid scrum kept the hosts at bay and allowed Parks to clear the danger.
The Scotland number 10 missed a second long range penalty - again he had the accuracy but not the required distance - and crossing between Evans and his centre partner Graeme Morrison brought a promising move to an end.
Neither side could maintain a foothold in the opposition's 22 in a hard-fought and frenetic finish to the first half. There were knock ons and plenty of turnovers as no quarter was asked or given.
There was some light at the end of the tunnel for the Ospreys. Coming up to the break, Bishop managed to block an attempted clearance kick from Parks but the centre frustratingly knocked on as he tried to gather the bouncing ball.
The disappointment of that missed opportunity was fully felt when Glasgow stormed out of the blocks in the second period, cutting the Ospreys' 7-0 half-time lead down to just two points.
Hooker Thomson failed to find his man at a 42nd minute lineout but Holah's loose knock down invited McMillan onto the ball. The Glasgow scrum half took it on before passing for Thomson to barge over in the left corner past a poor tackle from Ryan Jones.
Parks' conversion attempt faded away to the right and although he was able to produce some fine tactical kicking, Glasgow were left chasing the game by the hour mark.
A few knock ons aside, the Ospreys upped their game in the third quarter and Biggar's 51st-minute penalty, after good work by Phillips, steadied the men in black. And three minutes later, they got over for what was effectively the match-winning try.
Williams countered and set up a ruck near halfway, Hook picked and went down the blindside, handing off Glasgow captain Alastair Kellock on the way. The Ospreys centre then left-footed the ball around the covering Parks and won the race for the touchdown to finish off a terrific solo try.
Biggar's successful conversion pushed the Ospreys' advantage out to 17-5. Try as they might and they did - with Morrison, Beattie and replacements Hefin O'Hare and Colin Shaw desperately trying to unlock the Ospreys defence - Glasgow failed to get back in touch.
Flanker Collins, who typified the Ospreys' commitment, was left floored by an accidential knee to the head by Thomson. The New Zealander was replaced by Ian Gough and the retiring Filo Tiatia also got game-time as the Ospreys ground out a deserved win.
Helped by an effective lineout and an injection of pace by their backs, Glasgow, who have lit up this season's competition at times, continue to hunt for scores.
They won a kickable penalty late on, but elected to punt for a lineout with time not on their side and the attack eventually petered out with Williams clearing.
The visitors had the bulk of the possession in the closing stages but the Ospreys were content to soak up anything the Warriors could throw at them.
In the end, a late penalty from Biggar clinched his side's qualification for the final, with the Welsh region's first league crown since 2007 now just 80 minutes away. The result also means Scotland's wait for a first league winner goes on.
Speaking after the game, a relieved Shane Williams said: "The try was given, I know I was pretty close to the line but I'll take the five points.
"It's always tough against Glasgow. They didn't come here to lie down and they've been tremendous away from home all season.
"We made far too many mistakes in critical areas, but a win's a win."
|Ospreys Score Card|
|Glasgow Warriors Score Card|