The 2004 European Cup winners had Mark van Gisbergen to thank for rescuing them with his fifth successful penalty eight minutes into injury-time after an intense Causeway Stadium clash.
Wasps, though, must now win all four remaining pool games - including the return against Toulouse in France - to have any realistic chance of progressing.
Toulouse scored tries through full-back Gareth Thomas and wing Vincent Clerc, while Frederic Michalak dropped a goal and Jean-Baptiste Elissalde landed a conversion as the French giants trooped off in a far happier state of mind than their hosts.
A mistake-ridden performance by the Ospreys saw the Welsh region's hopes dented in the south of France as Clermont-Auvergne ran out 34-14 winners.
The Ospreys were shot back down to earth with a bump by countryman Stephen Jones but had themselves to blame for leaking four tries to their hosts.
The Ospreys got the start they were dreaming of when Shane Williams jumped to catch Shaun Connor's clever cross-field kick after just four minutes, but Connor missed the conversion.
Some weak tackling allowed centre Raphael Chanal to burst through the Ospreys ranks before his pass sent flanker Alexandre Audebert under the posts.
Jones, playing his first game after a month out with an ankle injury, converted the first of his perfect six from six.
Ospreys full-back Damien Karauna then failed to deal with nothing more than a speculative kick that handed Breyton Paulse their second.
Connor brought the Ospreys back into the tie with a drop goal on the stroke of half-time and then a second penalty three minutes into the second half.
The visitors' defensive frailties hurt them once again as Clermont flexed their muscles in the final 20 minutes.
The Ospreys finally buckled after concerted pressure from Clermont's forwards that ended with Gonzalo Longo-Elia driving over.
Replacement Anthony Flock then made a stunning break just moments after taking the field, captain Aurelien Rougerie kept the attack going before flanker Sam Broomhall went over to clinch a bonus point that leaves Pool Three wide open after the first two rounds.
Dan Parks was Glasgow Warriors' captain calamity as their cup hopes were shot to pieces by Leinster at Hughenden, in a 33-20 defeat.
The Scotland stand-off squandered 16 points' worth of goal attempts to hand Leinster the clear-cut victory they craved to keep alive their quest to squeeze into the quarter-finals.
While Parks was enduring a nightmare, his opposite number and rival skipper Felipe Contepomi underlined his world-class credentials by landing a series of crucial strikes and sliding in for the try which finally killed off the home challenge.