Millie Turner, Ella Toone, Lucy Staniforth and Amy Turner found the net in the first half before a double from Kirsty Hanson completed the scoring after the break.
United will now face Leicester City, recent champions of the FA Women's Championship, in the fifth round at Leigh Sports Village, next month.
Returning to club action after the international break, the Reds made a confident start, enjoying the majority of the possession in the early stages of the tie.
AfterStaniforth had fired over United’s first attempt of the contest, a good sliding block from Burnley’s Cara Bickett moments later stopped Sigsworth from opening the scoring from close range.