Blues dominated Championship opposition throughout and eventually made the pressure tell 10 minutes into the second half at the Proact Stadium, Chesterfield.
More space was afforded after that but the visitors could not take further advantage of many promising positions.
It means Blues will be one of 16 qualifiers in the hat for tomorrow’s (Monday’s) fifth round draw.
Manager Marta Tejedor made four changes from the 2-0 league defeat to Manchester City with Emma Kelly and Georgia Brougham handed full debuts.
Captain Kerys Harrop and Brianna Visalli also started with Sarah Mayling, Rebecca Holloway, Rachel Williams and Claudia Walker on the bench alongside the recently injured Connie Scofield.
Blues started brightly and gradually began to exert control through patient possession, creating the first chance in the 14th minute.
Scott, restored to her natural full-back position, ventured forward to take a pass from the left in her stride, stepping inside to find Abbi Grant at the second attempt only for the frontrunner to scuff into Emily Ramsey’s gloves from eight yards.
Blues turned the screw with Visalli jetting in from the right to withstand some grappling to cut back for Grant but the centre was a little heavy and pinged off target.
Lucy Staniforth picked out Grant on the left with Visalli forced wide from a shooting position by Alethea Paul before driving narrowly past the angle of post and bar at the near post.
More sprightly approach play down the right saw Scott feed Grant to drive in but this time Leandra Little was there to cut out at the near post.
Kelly surged down the left and chopped back inside to find Lucy Whipp whose drive was clutched by Ramsey seven minutes after the restart, while Staniforth bent a low ball to the far post that just evaded Kelly seconds later.
The breakthrough finally came as a result of a bit of ingenuity with Grant’s short corner whipped to the far post by Staniforth with Scott on hand to calmly guide her header inside the far post.
It might have been two in as many minutes when Staniforth arced to the far post with Chloe Dixon hurriedly clearing the danger.
The advantage was soon doubled, though, with Scott digging out a cutback from close to the line for Staniforth to rush into the box and fire a crisp effort inside the near post via the underside of the bar.
Grant was a fraction away from connecting with a low ball in front of goal and Visalli’s jinking run almost got her past the covering Naomi Hartley as Blues continued to probe.
Grant’s lift towards goal was covered by Dixon at the near post heading into the final 15 minutes, while substitute Mayling’s inswinging corner forced Ramsey into a hurried intervention.
Visalli brushed the outside of the post when attacking a looping ball on the run with nine minutes to go and Walker signalled her intent from the bench by cutting in from the left and swinging narrowly off target as a host of red shirts came at her on the edge of the box.
Mayling swung to the far post only for Walker to thump wide and Grant was thwarted one-on-one by Ramsey having been threaded down the left of the box by Chloe Arthur as Blues hunted more goals.
And Mayling would eventually get an assist of her own, lifting in for Scott to loop over Ramsey to add gloss to the score at the death.
Sheffield United: Ramsey, Paul (Sarri 59), Little, Hartley, Dixon, Cusack, Barker, Green, Fergusson (Lord-Mears 59), Wilkinson, Pennock.
Unused subs: Johnson, Batty (g/k).
Blues: Hampton, Jordan, Harrop (Mayling 71), Brougham, Scott, Whipp, Staniforth (Williams 71), Arthur, Grant, Visalli, Kelly (Walker 78).
Unused subs: Holloway, Brooks (g/k).
Goals: Scott 55, 90+2, Staniforth 59.
Referee: Adam Williamson.