Wasps battled back from 23-9 down to level the scores at 23-23 before being sunk by Yachvili who produced a 100% kicking display, landing four penalties and converting first-half tries by former England full-back Iain Balshaw and prop Eugene Van Standen.
Wasps replied with tries from winger Christian Wade and centre Ryan Davies, with fly-half Nick Robinson landing both conversions and three penalties.
Biarritz will be at home to the winners of Sunday's clash between French side Brive and Welsh region Scarlets in the semi-finals
Biarritz should have gone ahead in the fifth minute when fly-half Julien Peyrelongue and Balshaw split the Wasps defence but flanker Wenceslas Lauret fumbled the Englishman's final pass.
Wasps struck the first blow when Robinson kicked a simple opening penalty after 12 minutes.Balshaw was proving a real handful but, with speedster Takudzwa Ngwenya outside him, he cut inside and was halted by Hugo Southwell's crunching tackle.
Robinson doubled the Wasps lead eight minutes later with his second penalty following an inexplicable error by Balshaw, who was under no pressure when he deliberately knocked the ball into touch following a kick to the corner from Wasps scrum-half Charlie Davies.
It took Balshaw only a minute to make amends with the first try of the game. Robinson, under pressure from USA star Ngwenya, failed to take Yachvili's restart and the Zimbabwe-born winger slipped the ball to Balshaw who had a clear run to the line.
Yachvili added the conversion to edge Biarritz 7-6 ahead and, five minutes later, increased the lead with a penalty.
Lauret limped off after another knock-on but his departure did Wasps no favours with the visitors introducing France back-row star Imanol Harinordoquy from the bench.
Robinson's third penalty cut the deficit to one point three minutes before the interval, but the French again responded quickly.
The dangerous combination of Balshaw and Ngwenya linked up once more in a move which ended with prop Van Staden squeezing in at the right-hand corner flag for a try awarded by television match official Iain Ramage.
Yachvili landed the difficult conversion from the touchline.
The young Wasps, with six starters aged 21 or under, took the game to Biarritz in the second half before Yachvili's second penalty, earned as his forward forced the Wasps scrum into retreat, stretched the lead to 20-9 after 52 minutes and he added a third on the hour mark.
Wasps' efforts were rewarded with two tries in a three-minute spell.
After 62 minutes Robinson's through kick, deflected off a Biarritz player, bounced into the arms of Wade who marked his return from a three-month injury absence by racing in unchallenged.
Balshaw produced a try-saving tackle on Wasps replacement Jack Wallace but they struck again with Davies forcing his way over between the posts. Robinson converted both, and the sides were level at 23-23 but suffered the heartbreak of Yachvili's late winning kick.