Colin Nish's left-footed shot from outside the box was the closest Dumbarton came to the Gers' goal, while Richard Foster saw a similar effort fizz wide at the other end, letting fly from outside of the box and coming close, but not close enough to trouble Jamie Ewings.
The half looked as though it was going to fizzle out after a stoppage due to a Kenny Miller injury, but on the stroke of half time Kris Boyd broke the deadlock with a right-footed shot from the centre of the box to the bottom right corner, sending the visitors in on a high.
Rangers dominated the second period but, despite a number of efforts on goal, could not find a second goal to wrap the game up.
Boyd should have had a second when he found himself in on goal, but Ewings got out quickly to block with his legs. Miller, Lewis Macleod and Ian Black all had decent efforts but were unable to add to the lead.
The draw for the fourth round of the Scottish Cup takes place on Monday.