Centre-back pairing Callum Frogson and Jordan Haywood put them two up within the opening eight minutes before Josh Timson made it three just before the break.
Kiarn Nyemba, Wayde Hines and a brace from first-year full-back Harvey Ridgley sealed the win in the second half as they booked a date with the winners of Deeping Rangers or Leicester Road.
Hoping to repeat their exploits of last season - reaching the Third Round Proper against Premier League West Brom as the lowest ranked side in the competition - they went ahead when Frogson leapt highest to nod Hines’ corner past Ben Rose with only four minutes played.
Not to be outdone, and in a similar manner to the first, defensive partner Haywood – loitering with intent – pounced quickest to prod home United’s second moments later after another Hines corner caused panic in visiting ranks.
The dangerous Jericho Bonner then cracked an effort off the inside of the post after beating three Kimberley shirts before Hines had his own improvised shot cleared off the line as United dominated start to finish.
Welfare did eventually manage to test Ryan Camm in the home goal with two long range efforts but could yield very little as they succumbed to relentless pressure.
And they conceded a third just before the break as an entire team move led to Hines slipping in Timson who slotted home with ease.
Well ahead at the break and without entering second gear, Basford had all but sealed their next round status and added a fourth when Nyemba rounded Rose to tap home a goal his performance deserved.
But their best was to come when Ridgely, dancing forward with poise and flair, skipped around the advances of four before slotting United’s fifth past the isolated and powerless ‘keeper.
Rory Harrison was introduced with 21 minutes remaining, ensuring that no less eight players boasted some first-team experience under the watchful eye of Steve Chettle.
He was involved in their sixth when his pass found Hines who smartly lobbed over Rose from 20-yards for his customary inclusion on the scoresheet and Ridgley rounded things off when his shot deflected in via a post for his second of the night.
“The result was good. They’ve done the job and won the game, but we could play better, and we know that,’ manager Josh Law said. “Just in terms of our standards and how we go about it but that’s what this group comes to expect now.
“We demand high [standards] from them and if they drop below it, we’d like to let them know that they’ve got back to what they’re doing.
“I think they did that second half. We upped the tempo and created some good chances and scored some really good goals – young Harvey [Ridgely], two goals for a first year on a night he won’t forget.
“The aim is to try and get these boys around the first team as quickly as we can,” Law rounded off.
“Natty [Nathaniel Peters] misses out tonight because he’s been with the first team at Grantham on Monday which is absolutely brilliant, and we’ll take that if three or four can do that - that’s the aim of the club and that’s what we’re all trying to do.
“It all adds to occasions like this, and we hope to go on another run and take one game at a time. When those boys came back into it and you hear them now, they’re a bit more confident and a bit more vocal; ordering people about – long may that continue.”
“We’ll take anyone scoring,” coach Callum Chettle added on whether they emphasise the importance on several goal scorers.
“The two centre-halves, Callum [Frogson] and Jordan [Haywood], that have scored the first two, I don’t think have scored for two years almost.
"The wing-back has scored twice and Wayde [Hines] scored again, so any goals from any position on the pitch is always bonus for us.”
BASFORD UNITED: Ryan Camm, Harvey Ridgely, Louis Nicholson, Nassa Abdullah (Jamie Muir 45’), Callum Frogson, Jordan Haywood (C), Jericho Bonner (Rory Harrison 69’), Sam Newell, Kiarn Nyemba (Lewis Rourke 78’), Wayde Hines, Josh Timson
SUBS NOT USED: Tyler Dacres, Charlie Fox, Lui Bull, Gianni Musson (GK)
KIMBERLEY MINERS WELFARE (from): Ben Rose, Josh Beeston, Nathan Stevenson, Alfie Newbold, Josh Caley, Dan Earley, Tom Ellis, Alfie Hawksworth (C), Matthew Dickinson, Robert da Silva Makamu, Dan Walker, Lewie Sankey, Taylor Porter, Jeffreys Sagde, Dan Leonhardt, Theo Simmons
MATCH ATTENDANCE: 149