Ospreys produced a bonus-point 31-22 win over Wasps at the Ricoh Arena but Exeter's win at Bristol saw them fall short of the Anglo-Welsh Cup semi-finals.
Wasps failed to convert an early opportunity when their forwards turned the ball over under the visitors' posts, with this giving an indication of the pattern which dominated the first half.
The home forwards, including some experienced Aviva Premiership players, applied pressure in the scrum and through a well-coordinated driving maul, while Ospreys threatened with ball in hand.
A series of close-range set-pieces eventually saw home centre Brendan Macken open the scoring with a try which on-loan fly-half Warren Seals converted.
But Ospreys bounced back with a superb try, created by Dafydd Howells' back-of-the-hand offload to Jay Baker, who cruised to the corner.
Luke Price's conversion brought the scores level until Macken struck again, nine minutes before the interval, following another series of powerful thrusts from his pack.
However, it was the Welsh region who took a narrow two-point lead into the break thanks to replacement number eight Dan Baker's pick-and-drive score and Price's conversion.
A couple of thumping Alapati Leuia tackles then ended some early Ospreys second-half pressure, and won the field position from which Seals' penalty edged Wasps back in front.
Yet Ospreys again responded immediately through a simple Price three-pointer when Wasps were caught offside.
With a 20-match unbeaten run at the Ricoh to protect, Wasps introduced experienced forwards Jake Cooper-Woolley and Kearnan Myall in the second half, but despite maintaining an edge in the pack they still trailed by two points as the match passed the hour mark.
Home number eight Alex Rieder thought he had scored following a series of pick-and-drives, but referee Dean Richards adjudged him to be held up.
This did however give Wasps an attacking five-metre scrum from which a set move sent powerhouse centre Guy Armitage charging over for a try which Seals upgraded.
Incredibly, for a fourth time Wasps then conceded immediately after taking the lead. They failed to claim the restart, and after keeping possession through a number of phases, Ospreys sent replacement centre Kieron Fonotia over for a score which Price converted.
The Ospreys then turned over a home lineout to claim a bonus point try through Hanno Dirksen in the dying seconds, again converted by Price.