Leicester will need to beat four-time European champions Toulouse at Welford Road next Sunday in their quest for a prized Heineken Cup quarter-final place.
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The Tigers lie two points behind the French heavyweights in Pool Two following an absorbing Liberty Stadium tussle that saw Ospreys substitute Jonathan Spratt score a 77th-minute try to deny Leicester victory.
The visitors looked to have done enough through quickfire second-half touchdowns from scrum-half Ben Youngs and wing Niall Morris after they trailed 10-3 for much of the contest after Ospreys posted a 34th-minute Joe Bearman touchdown.
Dan Biggar and his opposite number, Leicester's Toby Flood, each kicked a conversion and penalty, but the Ospreys would have won had Biggar not missed four penalties - three of them either hit the post or bar - and a late conversion.
It was rough justice on the Ospreys, who are now out of the competition for another season, yet Leicester summoned all their trademark resilience to secure two priceless points away from home that sets up a mouthwatering encounter in seven days' time.
The Ospreys were without injured Wales internationals Alun-Wyn Jones, Ian Evans, Ashley Beck and Aaron Jarvis as they looked to put a huge dent in Leicester's quarter-final ambitions.
But the Tigers, who welcomed back a raft of England Test stars rested from starting duty against Aviva Premiership opponents Worcester eight days ago, started as marginal favourites.
The opening 10 minutes were close to Test match intensity as both teams pummelled each other at the breakdown and scrums, before an opening Biggar penalty attempt from 45 metres hit the post and bounced to safety.
And that was the story of Biggar's early fortunes, as he missed with two further penalties from similar range, one drifting narrowly wide and the other striking the bar.
On another day it could easily have been 9-0 to the Ospreys, yet Leicester could still make little headway and a Kahn Fotuali'i-inspired Ospreys breakaway cut open Leicester's defence.
It took a scything tackle by Tigers full-back Mathew Tait on Ospreys flanker Justin Tipuric to quell the danger, but further sustained Ospreys pressure resulted in another penalty chance that Biggar this time accepted.
Tipuric was again in the thick of things just two minutes later, placing a deft kick over Leicester's defensive line, and Tait escaped punishment despite appearing to challenge Tipuric late.
Such overwhelming possession and territory had to tell, though, and the Ospreys finally prospered six minutes before half-time.
Prop Ryan Bevington looked to have blown a gilt-edged chance when he ignored unmarked full-back Richard Fussell outside him, but Ospreys recycled the ball quickly and Bearman galloped over.
Biggar landed the conversion as Ospreys' former Wales captain Ryan Jones departed for treatment on a hand injury, which meant Lloyd Peers joining the action and James King taking up back-row duty alongside Tipuric and Bearman.
Flood kicked a penalty to open Tigers' account as half-time approached, but the visitors still had it all to do, trooping off 10-3 adrift.
Jones did not reappear for the second period, providing Wales with another possible injury concern just three weeks before they launch their RBS 6 Nations title defence against Ireland in Cardiff.
But the Ospreys battled on regardless, camping inside Leicester's 22 either side of a long-range kick and chase that almost resulted in a try for home wing Eli Walker.
Leicester had a more cohesive look to them than during the first 40 minutes, especially up front, and the Ospreys were not helped by a second injury blow that saw hooker Richard Hibbard depart clutching his ribs.
Tigers delivered an equalising try after outstanding work by the pack enabled Youngs to snipe his way over for a try that Flood converted and guaranteed a gripping finale.
Morris then struck with eight minutes remaining, but the Ospreys were not finished and an outstanding break from Walker caused enough mayhem in Leicester's defence for Spratt to finish off.
Biggar's conversion attempt, though, sailed agonisingly wide, leaving the Ospreys to reflect on a pool stage exit and Leicester focusing on Toulouse.