With both sides wearing similar blue-coloured strips, there was little to distinguish between the two but it was Cardiff who went into the interval with a 10-3 lead.
No. 8 Sam Feehan opened the scoring after a matter of minutes but after Jack Maynard added the extra two, Newbridge shored up their defence. Fly half Kieran Meek opened their account with Maynard replying with a penalty on the half time whistle.
The second half saw Cardiff step things up a gear as they stamped their authority on the game. Lock Miles Normandale and replacement prop Lewis Smout added further tries and with Maynard inch perfect from the tee, Cardiff progressed to the last eight.
With their pack outmuscling the Newbridge forwards early on, Cardiff camped themselves in the opposition half. Two huge scrums and a line-out surge sent the visitors back peddling with Feehan shunted over for the first try of the day. Maynard converted but for all their dominance, the home sided were just 7-3 ahead with a quarter of the game played.
Meek narrowed the deficit and missed another chance of three points as despite being under the cosh upfront, the Championship side stayed in the hunt. Cardiff continued to have more of the ball but with errors and indiscipline blighting their game, Newbridge were given repeated chances to clear.
The game then went into a period of stagnation, both sides losing centres to injury. Newbridge's Steffan Morgan followed Michael Stephens in limping from the pitch and although Cardiff continued to probe, wing Shaun Powell with one clean break, they had only a further Maynard penalty to show for their endeavours as the interval arrived.
Perhaps frustrated to go into the break just seven points ahead, Cardiff came out firing with Normandale the beneficiary of some sustained forward pressure. Maynard converted and added a 50th minute penalty and all of a sudden, their lead was a more comfortable 17.
Cardiff were now totally on top and with Newbridge conceding a succession of penalties, the Championship side were perhaps lucky not to go down to 14. Still, Maynard's third penalty extended the advantage and with the game now totally played out in the Newbridge half, the visitors had no way back.
With the game already over, the final 10 minutes did fizzle out but by that time, Cardiff had done enough - Smout's late converted score the final nail in the Newbridge coffin.
Cardiff: Tries: Feehan, Normandale, Smout; Convs: Maynard (3); Pens: Maynard (3).
Newbridge: Pen: Meek.