With Leicester loosing at Harlequins, the Devon side need maximum points from next weekend's trip to Worcester to achieve their goal.
Ospreys went into the break with a slender 10-8 lead after three tries were scored in a nine-minute spell with their touchdowns from Dafydd Howells and Morgan Allen, with Exeter replying through Luke Cowan-Dickie and a Ceri Sweeney penalty.
There were four tries in a nine-minute spell at the start of the second half with a second for England Under-20 player Cowan-Dickie and one from James Scaysbrook, both converted by Sweeney for a 22-10 lead.
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But Ospreys came back to secure a bonus point with a touchdown from Matthew Jenkins, converted by Matthew Morgan, and Tom Grabham to level the scores.
But a James Phillips try, converted by Sweeney, restored the Chiefs' lead.
But they were not content with Don Armand going over for their fifth try, converted by Sweeney, for a 14-point advantage.
The visitors attempted to reduce the arrears without success in the final minutes.
Both teams went into this game at the foot of their respective pools and with little chance of qualifying for the latter stages of the competition.
Exeter, with four players in the England Saxons squad at Gloucester and another two with the senior England squad, dug deep into their reserves and had Wales international fly-half Sweeney leading the side.
Ospreys were led by outside centre Jonathan Spratt as he made his 50th appearance for the Welsh region having made his debut in the 25-20 loss to Munster at Thomond Park back in November 2006.
The visitors were boosted by the news that their British & Irish Lions lock Alun Wyn Jones was bucking the trend of world-class players leaving Wales by signing a new three-year contract to keep him at the Liberty Stadium.
In addition, Ospreys have recruited hooker Sam Parry on a four-year deal from Newport Gwent Dragons. The 22-year-old is presently training with the Wales squad preparing for the forthcoming Six Nations campaign.
The visitors were the first to score, starting from a catch and drive which led to scrum-half Tom Habberfield giving a pop pass for wing Howells to touchdown but the cross-field wind, and swaying posts, made the conversion difficult.
The Chiefs countered with a similar move and it was hooker Cowan-Dickie who claimed the touchdown but Swenney's conversion from out wide on the left was blown across the face of the uprights.
Just before the midway point of the half Ospreys went back into the lead when Chiefs scrum-half Stuart Townsend misjudged the bounce of Habberfield's grubber kick. Wing Grabham was on hand to give the scoring pass to number eight Allen.
Chiefs were quickly back deep in the Ospreys half where openside flanker Arthur Ellis was sent for 10 minutes on the sidelines by referee Ben Whitehouse for killing the ball which allowed Sweeney to reduce the arrears from the kicking tee.
The home side, with a man advantage, tried to push Ospreys back 22 metres but first wing Fetu'u Vainikolo was halted with the line at his mercy and then Sweeney bundled into touch just short of line.
No sooner had Ospreys been restored to full strength when Cowan-Dickie was shown yellow for killing the ball after a Grabham break was brought to ground by full-back Tom James.
Exeter started the second half with clear intent as Ian Whitten broke but Vainikolo spilled the ball five metres from the line before Cowan-Dickie returned to the fray. After a couple of phases flanker Scaysbrook put his head down to cross the line.
Sweeney added the first conversion of the afternoon to give the Chiefs a 15-10 lead six minutes into the half.
The home side kept the pressure on with Cowan-Dickie driven over for his second try with Sweeney adding the extras.
Ospreys showed that they were still in the game as they tidied up a line-out with skipper Spratt putting centre Jenkins through for the visitors' third try. Fly-half Matthew Morgan's conversion reduced the arrears to five points.
Exeter's defence was stretched with Ospreys making an overlap out wide with Grabham scoring in the corner to secure a try-scoring bonus point and level the scores with 25 minutes still to play.
The home side replied quickly with their bonus point try coming through number eight Phillips which Sweeney converted from the touchline.
Howells then made a break down the left but Townsend was on hand with the try-saving tackle.
The Chiefs kept pressing Ospreys' line with blindside flanker Armand going over from close range for the Devon side's fifth try which Sweeney converted to complete the scoring.
Ospreys pile on the points to stay in European Champions Cup hunt
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15/04/2016 AT 21:01