But Mark Warburton’s team were made to work for the win against a stubborn League One side and, in Graeme Smith, an inspired goalkeeper.
After being frustrated by a series of impressive saves, the home side eventually made the breakthrough five minutes before half-time when David Templeton converted a low Barrie McKay cross.
Substitute Kenny Miller sealed Rangers’ place in the second round with a typical poacher’s goal, before right-back James Tavernier made the scoreline as one-sided as the game.
Rangers were dominant throughout, but Smith was comfortably equal to an early long-distance effort from Dean Shiels and a free-kick from Tavernier.
The ‘keeper was beaten midway through the first half when Tavernier whipped in a dangerous free-kick from the right, but Nicky Law’s header flew inches wide of the post.
Rangers came even closer to an opener moments later after a fine move started and finished with Lee Wallace, who forced another excellent low save from Smith.
But five minutes before half-time, Rangers finally got the goal their dominance deserved, as McKay controlled neatly on the left before delivering a low cross to Templeton who squeezed the ball into the bottom-right corner.
Rangers almost doubled their lead just three minutes into the second half, but Peterhead’s Ryan Strachan was on the line to block a certain goal from a deflected Martyn Waghorn long-range strike.
The pattern of the first half continued into the second, with Rangers controlling the ball and creating plenty of chances, and Smith continuing to frustrate them with a string of fine saves.
But substitute Miller made things more comfortable 15 minutes before the final whistle.
The impressive Tavernier, one of manager Warburton’s new recruits, created the goal, driving into the box and cutting the ball back to Clark who failed to make contact. But Miller was on hand to finish at the back post.
Tavernier himself added some gloss to the scoreline with 10 minutes remaining, running onto a through ball from Clark and rattling a low shot beyond Smith to make it two goals in as many games for the defender.
There was still time for Smith to make one more superb close-range save from Waghorn in a fitting finish to the game.