Striker Shaun Miller netted a brace, his first professional goals in United colours whilst defender Harry Maguire too netted twice to fire the visitors into a four-nil lead before former Blade Jordan Stewart netted a scant consolation for the home side.
Danny Wilson had made changes to the side which drew 1-1 with Oldham Athletic last Saturday, Darryl Westlake (for Tony McMahon), Shun Miller (for Dave Kitson), Matty Harriott (for Kevin McDonald), Jordan Chapell (for John Cofie) and Chris Porter (for Nick Blackman) were all included in the starting eleven, whilst it was a welcome return for defender Lecsinel Jean-Francois who has recovered from injury.
Little opportunities of goalscoring were created in the opening exchanges, and the Blades first attack of any mention came when Miller turned his marker; Bencherif to win a vorner but Flynn's deliver was cleared by former Sheffield Wednesday striker Enoch Showunmi - and at the other end, following Arquin's cross, Jeff Hughes headed wide.
Since Mark Howard's injury at Leyton Orient 10 days ago George Long has proved an excellent replacement and the youngster was called into action after 22 minutes, Showunmi saw his header saved after Sheehan produced a free-kick, it was won when Westlake fouled Zoko. United were soon in the lead when Westlake was in the thick of things; he found Porter on the right and his excellent ball to the unmarked Miller was volleyed into the goal; 0-1.
It was the first time the pair had started since the Capital One Cup defeat to Burton Albion and the dup looked a dangerous pair, but Long again produced a timely save to keep the home side at bay, he flicked over from Hughes header after he towered over Westlake.
As the first half drew to a close United had one final say and doubled there lead in a mess-up in the box. Miller, on the right hand side crossed, keeper Bialkowski got down to tip Flynn's drive around the post. From the resulting corner Maguire took advantage of a thumbing Arquin to drill the ball home from close range; 0-2.
At the re-start of the match the home outfit made their intent by the introduction of Lee Hughes, who scored at the Lane a few weeks previous and former Barnsley and Rotherham United winger Jamal Campbell-Ryce to replace Mahona and Showunmi but it was United who had a sight of goal when captain Doyle fired wide.
The Blades pressed forward, in search of a third and were soon rewarded for their efforts, on 52nd minute when Miller rifled home his second goal of the match, finishing from Flynn's corner. Two minutes later it was 4-0 which was netted in odd circumstances, Maguire picked up a loose ball on the right wing and his ball into the box caught out number one stopper Bialkowski and flew into the net with the aid of the woodwork; 0-4.
It was a great night for United and for the fans who had doubts prior to the match, but they didn't let the loyal supporters down, with the only negativity of conceding a goal. But, with 23 minutes remaining the hosts missed a great opportunity to score, with Jeff Hughes failed to net with the goal at his mercy, but at the other end Westlake was denied a fifth when Liddle blocked his drive.
The Meadow Lane club were eventually rewarded with there efforts but it was little too late and were sure that boss Keith Curle will only be thankful that three points weren't at stake in this match. Former Blade Stewart (75th minute) nabbed the ball from Hill before producing a curling left foot hit with Long given no chance to save; 1-4. Jean-Francois replaced Chapell to make his first start since March, he received a good applause from the traveling 483 Blades supporters.
County searched for another goal towards the end of the match, Lee Hughes missed from close range and Joe Ironside came on to wind down the clock as the referee blew time on the game and United marched onto round three were they will face Coventry City.