The Blues ran out 20-6 winners at Edinburgh, condemning the Scottish franchise to a second successive quarter-final defeat in the competition.
A Jaco van der Walt penalty put Edinburgh ahead after 19 minutes, but Cardiff was in control by half-time thanks to converted tries from Ellis Jenkins and Blaine Scully.
Although Van Der Walt briefly narrowed the hosts' deficit, there was to be no fightback and two Jarrod Evans penalties ensured the Blues, for whom Josh Navidi was outstanding, triumphed with a degree of comfort.
Next up for Cardiff is a home clash with Pau in the last four, while Gloucester, a two-time Challenge Cup winner and last year's runner-up, will host Newcastle Falcons after beating Connacht 33-28.
Henry Trinder's solo score was the pick of Gloucester's five tries in Galway, where a late Connacht rally proved in vain.