Carnegie topped the north group with five wins from their six group matches, two of those victories coming against Yorkshire rivals Doncaster Knights.
Their impressive progression has set up a quarter-final clash with Bedford sometime over the weekend of February 1-3.
Bedford won three and lost three in their group stage, finishing third behind London Irish and Cornish Pirates. They currently occupy fifth spot in the Championship, 23 points clear of bottom-placed Carnegie and just nine points adrift of top spot.
The Cup has proved a welcome distraction for Carnegie after a dismal start to the Championship, which has seen them win just one of their opening nine matches to leave them four points adrift of second-bottom Hartpury College.
Elsewhere in the competition, league leaders London Irish host Coventry whilst fellow capital club and second-placed side Ealing Trailfinders welcome Nottingham. Also vying for a place in the final four, Cornish Pirates travel to Jersey Reds.