With the Blades third in the table and the Bluebirds battling to avoid relegation, it was no surprise that both teams made seven changes as they prioritised their league efforts after a busy festive schedule.
Kion Etete came into Cardiff's line-up to make his first appearance of the season after a hamstring injury, and the striker showed some understandable signs of rust as he squandered two excellent chances in the opening six minutes.
The visitors kept pressing forward and got their reward with an opening goal that was all Ashford's doing as the 20-year-old academy product stole the ball off a dithering Norrington-Davies and hared into the penalty area before coolly finishing past Adam Davies.
United had ample possession, but struggled to create proper chances, with their only shots on target in the first half fired straight at Cardiff goalkeeper Ethan Horvath.
Chris Wilder's men started the second half at a quicker tempo and with greater attacking intent as Ryan One and Rhian Brewster forced Horvath into action.
Their opponents still carried a threat on the break through the likes of Ollie Tanner and Ashford, who almost scored a second as his powerful 20-yard shot was pushed away by Davies.
With three minutes left, Burrows looked to have conjured the moment of quality that United had been so sorely lacking when he curled a fine effort from the edge of the penalty area which beat Horvath, but cannoned off the far post.
Cardiff, FA Cup finalists in 2008, survived that brief scare and can now look forward to playing in the fourth round, with ties to be played on the weekend of 8 February.