The Tigers' 52-0 bonus-point hammering of Treviso leaves them four points above the Welsh region at the top of pool three ahead of next week's group decider at the Liberty Stadium.
Leicester will go through as group winners if they win, draw or claim two losing bonus points.
The Ospreys will overtake the Tigers if they win and deny them a bonus point, or if they claim a bonus-point win and Leicester only get one bonus point in defeat.
If the Welsh side win but Leicester top the group, the Ospreys could still advance as one of the two best group runners up.
The visitors emerged to ringing boos at the Stade Aime Giral, known as the Bull Ring, the volatile Catalan crowd still fervent following Perpignan's controversial 15-9 defeat in Swansea last October.
The Ospreys started promisingly, looking to stretch the Perpignan defence out wide.
Byrne will be a major injury worry for Wales and the Ospreys
A Tommy Bowe hack through nearly led to a try, the Ireland wing just beaten by the bounce, but every other half-break was followed by spilt possession and forward passes.
The home side, in contrast, were clinical, fly-half Meyer making the most of their rare first-half visits into Ospreys territory.
The stand-in for the injured Dan Carter landed penalties following indiscipline by Marty Holah, Filo Tiatia and Hook, the Ospreys repeatedly warned by referee Alain Rolland.
In-form full-back Byrne hobbled off with an ankle injury after colliding with Phillips as he went for a high ball, creating a major worry for the Ospreys and Wales.
After the unfortunate clash, Phillips then transgressed following another strong Perpignan surge, the scrum-half sin-binned as Meyer's penalty made it 12-0 at the break.
The home side upped the pressure at the start of the second period, the Ospreys barely hanging on until Phillips' return.
Perpignan continued to go for the jugular, though, booting a kickable penalty to touch.
Their formidable pack rumbled to the Ospreys' line, Tuilagi finally battering through for a try from close range.
The Ospreys at last delivered on their attacking potential as Phillips powered into the Perpignan 22 with a dynamic run.
WELSH HEINEKEN WINS IN FRANCE
2008/9: Biarritz 6-10 Blues
2007/8: Bourgoin 21-28 Ospreys
2006/7: Toulouse 34-41 Scarlets
2006/7: Bourgoin 5-13 Blues
2000/1: Colomiers 6-19 Llanelli
1999/2000: Bourgoin 30-36 Llanelli
The ball was quickly recycled and spread right, Williams gathering an awkwardly bouncing ball, spinning out of Tuilagi's tackle, and sprinting home for a typically dazzling score.
An overlap was created for Alun Wyn Jones, who cantered over the line only to be called back for Tiatia's careless forward pass.
The visitors' pressure continued, and Bowe showed the strength to shrug off a tackle in midfield, draw the cover and send replacement Vaughton over for a converted try.
Hook's only penalty cut the deficit to two points, but, with just four minutes remaining, the Ospreys could not break Perpignan's remarkable 15-game home winning streak in Europe.
Ospreys coach Sean Holley: "We came with a positive frame of mind to go and get the win, but in the first half we weren't clinical enough and it cost us in the end.
"They got away from us and it's hard to figure out how because they didn't really create that much, they just lived off our errors.
"That's something we have to sort out for next week because Leicester are very similar, they will take their chances.
"Looking at the positives - the way we came back in the second half, I think another five minutes would have told."
Perpignan scrum-half Chris Cusiter: "We deserved to win the game, after Henry Tuilagi scored we were in control.
"We made more of the possession and had taken advantage of it when they killed the ball by kicking the points.
"We didn't make enough of the period when Mike Phillips was off and full credit to the Ospreys for coming back strongly.
"It was a high level of intensity game and pretty fast in the first half."
Perpignan: Burger, Manas, Marty, Mermoz, Candelon, Meyer, Cusiter, Pulu, Guirado, Mas, Britz, Olibeau, Le Corvec, Chouly, Tuilagi. Replacements: Hume for Meyer (66), Durand for Cusiter (66), Chobet for Pulu (47), Geli for Olibeau (76), Perez for Le Corvec (47). Not Used: Kairelis, Porical.
Ospreys: Byrne, Bowe, A. Bishop, Henson, Williams, Hook, M. Phillips, James, Hibbard, A. Jones, Gough, A. Jones, R. Jones, Holah, Tiatia. Replacements: Vaughton for Byrne (37), Bennett for Hibbard (52). Not Used: Griffiths, Bateman, Smith, Nutbrown, Biggar.
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