Rovers, who reached the final of the prestigious competition last season, triumphed 3-2 on penalties after the first round fixture finished 1-1 at the end of normal time.
The visitors opened the scoring on 34 minutes when Keira Walsh played in Aisha Yallop, who coolly chipped the keeper, but North Yorkshire levelled 20 minutes later.
Rovers dominated possession for long periods, but just couldn’t unlock the North Yorkshire defence.
Their quality and experience shone through in the shoot-out, with Rovers keeper Megan Pope making a crucial save.
ROVERS: Pope, Hulton, Godfrey, Samuels, Mallett, Taylor, Walsh, Holland, Yallop, Lamb, Doze.
Subs: Maher, Johnson, Fenton, Chandarana.