Full back Ben Blair helped himself to 25 points as he added the extra points to all seven tries, bagged one himself and also landed two penalties. There were also two tries for Tongan prop Tau Filise.
The Falcons defended bravely in the opening 40minutes and should have been within one point of their visitors at the break. But acting skipper Jimmy Gopperth missed with a 45 metre penalty on the stroke of half-time and that allowed the Blues to go into the interval with a 17-13 lead.
The only Welsh side now left in either European competition, the Blues ran in two tries in the opening 15 minutes to lead 14-3, Casey Laulala and Martyn Williams crossing, but were pegged back to a single point advantage when Charlie Amesbury slipped out of a tackle to cross.
Gopperth added the conversion and also kicked two penalties, but then fatally missed with his last penalty shot of the half.
The Blues were in no mood to hand around in the second half to secure the win and added 31 points in a blistering 18 minutes that ruthlessly killed off their hosts' hopes. Filise charged over for the first and last of the four tries they scored in that time and Blair and the outstanding Jamie Roberts got the others.
Blair couldn't miss with his golden boot and kept the scoreboard ticking over. He added the extras to all four and then also converted Ceri Sweeney's final try.
Victory for the Blues means they are the only one of the three Heineken Cup teams who joined the Amlin Challenge Cup at the quarter-final stages who survived.