The Gers went behind to a Dylan Lawrence strike on the 14th minute, but goals from Kane Ritchie-Hosler, Aaron Lyall, Chris McKee and Murray Miller ensured they booked their place in the next round.
Rangers had a chance early on as Robbie Fraser’s cross found Cole McKinnon just inside the box. The midfielder got a shot away at goal but it was touched wide by Ross County goalkeeper Jamie Stephen.
Kane Ritchie-Hosler was next to have success down the left as he again found the forward run of McKinnon but this time his shot was just over the target.
It was the home side who made the breakthrough on the 14th minute as the ball broke for Matthew Wright who round Dylan Lawrence in space in the box and the striker fired a powerful shot beyond Jay Hogarth.
Rangers pressed for an equaliser as Kane Ritchie-Hosler crossed into the box with Murray Miller’s shot blocked by Ross County. The rebound fell to striker Chris McKee who diverted the ball into the back of the next but it was flagged for offside.
On the 29th minute Rangers broke from a Ross County corner as Kyle McClelland headed the set-piece clear before Alex Lowry fed the ball through to Robbie Fraser. The Gers full back powered towards goal, but his shot was just wide of the target.
The Gers were back level on the 34th minute as Kane Ritchie-Hosler cut in from the left before drilling a low shot across goal and into the back of the net for his third goal in three games.
David McCallum’s side were ahead with three minutes of the first-half remaining as Alex Lowry played the ball through to Aaron Lyall who shot beyond Jamie Stephen.
Rangers added a third early in the second-half as Robbie Fraser’s cross was headed home by Chris McKee.
It was almost a fourth shorty after as Fraser again provided the assist for McKee with another delivery into the box but the strikers’ shot was saved by Jamie Stephen.
The Gers were in control of the cup tie now as they came close again as Alex Lowry’s corner was spilled by the keeper and fell to Harris O’Connor but his touch was just wide of the target.
The fourth did come on the 73rd minute as midfielder Murray Miller found space on the edge of the box to curl a right-footed effort into the back of the net.
Jay Hogarth was called into action a few minutes later as he made a good save to block Josh Reid’s shot and deny Ross County a second of the evening.
Substitute Ross McCausland almost added a fifth in the last few minutes of the game as he worked his way into the box before taking a shot at goal but he was denied by the post.
Rangers: Hogarth, O’Connor, Fraser (Henderson, 55), Miller, MacKinnon, McClelland, Lyall, McKinnon, McKee (McCausland, 74), Lowry (Anderson, 66), Ritchie-Hosler. Subs: Pazikas, Muir, Balde, Yates.
Ross County: Stephen, Brady, Reid, Williamson, Kane, Riddell, C. Mackenzie, Mackinnon, Wright, Graham, Lawrence. Subs: R. Mackenzie, Henderson, Ross.