Boyd struck on the stroke of half-time for the visitors and they held onto the lead in the second half against fellow Championship side Dumbarton.
The home side started brightly and had the game’s first chance when Mitch Megginson forced Rangers goalkeeper Lee Robinson to palm his low drive around the post.
Colin Nish came close to opening the scoring for Dumbarton on 19 minutes but his left-footed strike from the edge of the box flew just wide.
Boyd had the ball in the net for the visitors soon on 42 minutes when he deflected Richard Foster’s long-range shot into the net but the striker was ruled offside.
Boyd did put Rangers in front on the stroke of half-time when he coolly slotted the ball into the bottom corner after Lewis Macleod threaded a pass through to him.
Boyd came close to grabbing a second on 50 minutes when he beat two defenders to get through on goal but Jamie Ewings saved well.
Kenny Miller did well to find space in the box minutes later and unleashed a shot that seemed destined for the net but Mark McLaughlin did well to throw himself in front of it.
Macleod threatened just before the hour mark when he expertly controlled a long raking pass on the right and cut inside before unleashing a left-footed shot that found the side netting.
The home side almost equalised on 76 minutes when Andy Graham’s downward header forced Robinson to make save to deny him.
Jon Daly had a great chance to wrap up the game late on when he got through one-on-one with Ewings but he blazed the ball over the bar.
Rangers survived some late pressure from the home side to ensure their name will be in the hat for Monday’s fourth-round draw at Hampden.