Young said: "We pulled out a big performance when it really mattered. Not many people thought we could come here and dominate the way we did but we showed just how far we have come over recent years to achieve something special.
"We tend to rip ourselves to shreds in Wales where it's always doom and gloom but having two Welsh sides in the quarter-finals proves the regional structures are working. This was a big win for us."
Victory saw the Blues top the pool but their failure to take a bonus point, or win by a 24-point margin, meant they failed to secure a home tie in the last eight and must travel to three-time European champions Toulouse.
Young said: "The priority was to get a win but once we built up a lead we tried to go for the 24-point deficit, or a bonus point, to get the home draw but Bristol made it very difficult for us. We're just happy to have qualified.
"No one will give us much chance in Toulouse but we're not in the quarter-finals to make up the numbers. We've proved we can go away and win tough matches."