The triumph was bittersweet for the Blues though as their failure to claim the crucial third converted try means that they now have the daunting prospect of trip to Toulouse in the last eight rather than a home tie against the three times champions.
In a first half full of incident, it was the home side who looked to have drawn first blood when, from an incisive run of his blindside wing, Anthony Elliot picked out Luke Arscott's acute line to put the young fullback over after only six minutes.
However, after initially awarding the try referee Alan Rolland consulted with his touch judge who confirmed the final pass was forward.
It was all Cardiff Blues after that as they attacked purposefully from depth and twice they had tries ruled out, firstly for Jamie Roberts for a forward pass and then for Martyn Williams whose foot was adjudged to have touched the line before he dabbed down in the corner.
The Blues were not to be denied though and with only a minute remaining before the interval, Molitika pounced on a loose ball a from a Bristol five-metre scrum and barrelled over to break the deadlock.
Blair converted on the stroke of forty minutes but the Blues weren't finished there. After a monster touch-finder from Nick Macleod had put the Blues back in range, Jason Spice spotted Gareth Thomas's decisive run around the blindside and the former Wales captain shrugged off the challenge of Sean Hohneck to put his side firmly in charge.
Blair stroked a beautiful touchline conversion and Cardiff went in 14-0 to the good.
The second half was short on the excitement of the first, and as the energy levels of the sides dipped, so did the quality of the contest. Both sides showed ambition but their failure to retain possession for any length of time, combined with a littering of handling errors meant the game lost its spark.
Blair registered the only points of the second period with a penalty for repeated Bristol infringing at the breakdown on fifty three minutes and for there on in, it was a story of the Blues' hunt for the elusive extra seven points.
They probed and foraged manfully in the dying stages, but Bristol held firm and now the Blues must gear themselves for April's monumental clash in the South of France.