The Blues squeezed past the 2006 finalists thanks to the only try of the game by Jamie Robinson, five points from the boot of Ben Blair and a sturdy defensive display.
The victory was the fourth in a row for Cardiff who top Pool Six by four points ahead of Gloucester and Young said: "We're not going to start talking about the quarter-finals with two big games remaining, but we are very pleased with this result.
"I'm proud of the character the boys showed to pull off this win. We knew this was never going to be easy in France and the conditions meant it wouldn't be pretty.
"But this was all about who wanted it most and which team was prepared to put their bodies on the line for the win.
"We showed real character and determination with our defence and deserved the win.
We started very well with the ten points but we had to dig deep in the second half as they threw everything at us."
History was stacked against the Blues given that Biarritz had only twice been beaten at Parc des Sports Aguilera in the Heineken Cup.
But they made a dream start when Blair kicked a tenth-minute penalty before Robinson was first to touch down after Gareth Thomas charged down a clearance kick by Biarritz full-back Benjamin Thiery.
Damien Traille and Julien Peyrelongue responded before half time but the outside-half also struck the post with a second effort before a tight second-half failed to produce a single point.
Young said: "The idea was to pin them back but the execution was off . We must have broke the world record for the number of kicks that were charged down. That was disappointing, but I was so pleased with the way we dealt with each mistake."
The defeat was the sixth in a row for Biarritz in all competitions and with just two rounds of pool games remaining, effectively ends their European campaign before the New Year.
Coach John Isaac said: "We're very disappointed with the result. We had most of the ball and territory but just like last week against Cardiff, we did not make it count on the scoreboard."