A thrilling injury-time try from scrum-half Harry Ellis gave Leicester the bonus point which keeps his side in the competition.
HEINEKEN CUP Pool One
Leicester (0) 17:
Tries: A Tuilagi, Ellis
Cons: Goode 2
Biarritz: (18) 21:
Tries: Thion, Marlu
Pens: Yachvili 2
Biarritz had a superb first half and went 18-0 ahead with the help of tries from Jerome Thion and Jimmy Marlu.
But with Wasps beating Calvisano, Leicester needed to claim at least a point to keep their Cup hopes alive.
They piled on the pressure and looked to be heading out but late tries from Alex Tuilagi and Ellis saved the day.
The match was a superb contest and continued the run of nail-biters that his tight pool has thrown up.
The Tigers had no answer to Biarritz's dominance and clinical error-free rugby in the first half.
The French side crossed after just five minutes when Thion came up with the ball after a drive from a close-range lineout.
Leicester could not get into their stride as their passing let them down and Biarritz, with the help of some superb tactical kicking, were able to exert all the pressure.
A penalty from Dimitri Yachvili and a drop goal from Damien Traille extended Biarritz's lead before Welford Road was stunned into silence when Marlu barged his way over.
Leicester's dreadful half was summed up in the closing stages when they earned a penalty after a rare foray into the opposing 22.
Skipper Martin Johnson did not go for the three points but fly-half Goode, who has been in such good form this season, inexplicably fired the ball out of play in the touch and goal area.
The second half started much better for the home side and Goode made up for his error with a penalty on 43 minutes.
Leicester knew they had to go for the bonus point to ensure their Heineken Cup survival but their poor handling at vital times let them down.
Ollie Smith was the prime culprit and after a superb break, he spilled the ball with the try-line in sight.
With the drama and tension rising, cool heads were needed but Ellis, who is vying for a place in England's Six Nations team, was sin-binned for a petulant elbow into the neck of Yachvili.
The French star then dusted himself down to struck the resulting long-range penalty on 55 minutes and further rub salt into Leicester's wounds.
But the Tigers are a resilient bunch and for the rest of the game Leicester laid siege to the Biarritz line.
The French side were simply awesome in defence and time after time they came up with try-saving tackles with the ubiquitous Serge Betsen leading the way.
Leicester did eventually break through the line through Tuilagi with four minutes to go to set up a thrilling finale.
With the volume rising Ellis dashed over and although his grounding was not certain, the video referee confirmed the try to delight the home crowd.
Afterwards, Leicester coach John Wells was unhappy as his side now have to rely on rivals Wasps to do them a favour in Biarritz next week if they are to reach the last eight.
"We've got to hope the team we have beaten twice can do us a favour," said Wells.
"It's not beyond the realms of possibility. We're desperately disappointed, even more so because we've got to wish Wasps well.
"A lot of things didn't go for us but we were responsible for our owndownfall. Biarritz strangled us and didn't let us play."
Biarritz skipper Thomas Lievremont added: "We showed a great spirit and a strong heart and tackled, tackled, tackled."
Leicester: Murphy; Smith, Lloyd, Gibson, Healey; Goode, Ellis; Morris, Chuter, White, M Johnson (capt), L Deacon, Moody, Back, W Johnson.
Replacements: Buckland, Rowntree, Kay, Abraham, Bemand, A Tuilagi, Vesty.
Biarritz: Brusque; Bidabe, Arramburu, Lacroix, Marlu; Traille, Yachvili; Balan, August, Avril, Thion, Olibeau, Betsen, Lievremont (capt), Harinordoquy.
Replacements: Gonzales, Lecouls, Lealamanua, Milheres, Couzinet, Dambielle, Dupuy.
Referee: A Lewis (Ireland)