Nichol Forster netted the only goal of the game on 17 minutes, finishing off a well-worked move, as a revitalised Rovers made it four wins on the spin in all competitions.
On an appalling pitch, Rovers started strongly, dominating possession and moving the ball quickly from side to side.
The breakthrough arrived just past the quarter-hour mark, when Amie Fleming picked up possession close to the halfway line, before playing a superb through ball to Forster, who rounded the keeper and fired home.
Rovers restricted the hosts to very few entries into their own half, whilst creating countless half-chances at the other end.
Amy Bland switched play from left to right, picking out Chloe Adams, whose near-post shot struck the side-netting. Rovers also threatened from corners, with Vicky Westerman going close on a couple of occasions.
Roared on by a vocal home crowd, Whitley Bay improved after the break, putting Rovers under pressure, but the visitors dealt with it well and still enjoyed plenty of possession.
Rovers again created good opportunities in the final third, but failed to find the killer pass in and around the box.
Sophie Flynn came closest to doubling Rovers’ advantage, but her shot from the edge of the box forced a superb save from the Whitley keeper, who tipped the ball over the bar.
Assistant manager Amanda Goodwin said: “We had a long journey over there and played on a boggy pitch, which the players found hard to run on, as they frequently lost their footing.
“We just told them to go out and get the job done, and the girls rose to the challenge. They worked hard and there is a good togetherness within the group at the moment. The scoreline didn’t reflect the performance, but we are delighted to get through to the next round.”
ROVERS: Jackson, Pennington, Seed, S. Hardy, Fleming, Westerman (H. Forster 46), Savage, Flynn, Adams, Bland (Musker 65), N. Forster.
Subs not used: Noori.