Welbeck were comfortably dispatched by 86 runs on Saturday before Kimberley produced a fabulous four-wicket win in the west midlands at Smethwick, who had knocked out holders Cuckney in the last round.
They now face Liverpool-based Northern in the semi-finals at home on August 18.
Opener Akhil Patel was the star of the show with a century against Welbeck and then a half-century against Smethwick - but he was backed up throughout by two fine team efforts.
Kimberley captain Dom Brown said: “That was a brilliant weekend for us.
“It was a close game but when you get to the last eight of any competition they are all going to be good sides.
“They had a lot of depth and are used to playing on their own ground, so we had to get used to our surroundings pretty quickly.
“The outfield was really big and really fast and the wicket a bit slow, but we adapted quickly and managed to win the game.
“The lads played well and Akhil Patel and Ximus Du Plooy were excellent.
"Akhil just seemed to run out of steam a little bit (against Smethwick) but Ximus then took us over the line really well.”
Choosing to bat against Welbeck, Kimberley amassed a daunting 276-5 in their 50 overs with Patel hitting a superb 113 with 13 fours and three sixes and sharing an opening stand of 98 with George Bacon (44).
Martin Weightman also hit 42 not out.
The visitors were always well behind the rate and were all out for 190 with 20 balls still to be bowled.
Opener Ryan Terblanche struck a fine 71 and Oliver Dyson added 33, but no-one else stuck around as Alex King (3-37) and Du Plooy (3-38) tore through them.
On Sunday, Kimberley asked the home side to bat first and they made 209-6 in their 40 overs, Rawait Khan with 56, Rapjal Beniwal 50 and RBS Venkataraman 37, while George Scottorn claimed 3-47 for Kimberley.
In reply, Kimberley reached their target of 210-6 with eight balls to spare, Patel again immaculate with 86, while Du Plooy ended on 59 not out.
Brown said: “Akhil had a fantastic weekend and we are all really pleased for him.
“He played a massive part in both the wins.”
This Saturday, Kimberley are away at Plumtree as they continue to try to bridge the 14-point gap to leaders Cavaliers & Carrington.
Brown added: “We have to forget the cup for now.
"We have Plumtree away this Saturday which is always tricky and that is exactly where our attention will be.”