Three goals in eight second half minutes enabled United to book a place in the second round of the Carabao League Cup.
Walsall scored first and last but an own goal, following a Ched Evans shot and goals from Nathan Thomas and Daniel Lafferty secured United's passage.
Blades boss Wilder changed the whole starting XI for the encounter. Wright captained the team and Brayford started. New boys Lundstram and Thomas made their first competitive starts, as did 'keeper Eastwood and defender Bennett, who was signed from Mickleover Sports last season. Evans was on the bench with six of the starters from Saturday's victory over Brentford.
In a slow start, there was little to report in terms of goalscoring opportunities early on. The first chance, on 11 minutes, saw United fail to clear twice before Lundstram blocked an effort from Morris.
Walsall did, however, open the scoring with their next chance, as United complained about offside. From Leahy's pass, Oztumer looked beyond the defence and from his centred ball BAKAYOKO sidefooted home.
United went close to levelling within minutes. A free-kick routine was initially repelled but Duffy's next cross was headed down by Hanson, Brayford's shot was blocked and then Lavery was crowded out.
Chances were hard to find for both sides but the Blades had another good opportunity before the interval. Brayford's pass put Lavery around the back but his cut back evaded Hanson as well as the visiting goalkeeper and a defender.
Both sides had a good chance each in the first five minutes of the second period. Visiting goalkeeper Gillespie flapped at a Duffy cross but recovered to save from Lavery and then, at the other end, a powerful volley from Edwards looked to be heading towards goal but Wright headed over his own crossbar.
Evans replaced the injured Hanson and soon after Basham took over from Bennett.
Just after the hour Lundstram's shot was deflected wide after he linked up with Brooks and Evans and the Blades were now forcing the visitors back.
A leveller seemed likely and it arrived on 74 minutes. Lafferty's header saw Thomas cross and from an Evans half-volley, goalkeeper Gillespie touched the ball onto defender ROBERTS before the home fans celebrated.
And it was quickly followed by a second goal - and then a third.
On 79 minutes, Evans was again involved, pulling the ball back from the byline for THOMAS to sidefoot home in a crowded box.
The third followed on 82 minutes. Duffy's free-kick was cleared to the edge of the box where LAFFERTY was waiting to fire back, beating Gillespie with the aid of a deflection.
United turned the game in the space of eight minutes and were now in full control.
Walsall, though, were handed a lifeline late on when Kouhyar's powerful shot was blocked by the arm of Basham. A penalty was awarded and OZTUMER confidently dispatached just as four minutes of added time was announced.
The extra moments passed with little drama, despite Walsall's urgency, as United's booked a round two place.
Blades: Eastwood, Lundstram, Thomas, Wright, Hanson (Evans 52), Duffy, Lavery (Fleck 86), Lafferty, Brayford, Bennett (Basham 58), Brooks.
Unused: Stevens, Clarke, Coutts, Blackman.
Walsall: Gillespie, Edwards, Leahy, Guthrie, Devlin, Chambers, Roberts, Oztumer, Morris, Kinsella (Kouhyar 83), Bakayoko (Jackson 74).
Unused: Flanagan, Sangha, Cockerill-Mollett, Candlin.