Kris Boyd enjoyed the perfect debut for Rangers as his second-half hat-trick helped his new team into the fourth round of the Scottish Cup.
Rangers only had one goal to show for their first-half dominance - Soti Kyrgiakos tapping in from close range.
But Boyd scored a penalty just after the break after Peter Lovenkrands was fouled, tapped in a second and completed the hat-trick with a header.
Ross McCormack rounded the victory off with the aid of a deflection.
Kris Boyd's debut almost got off to the perfect start when he came within inches of finding the back of the net seconds after kick-off.
Chris Burke embarked upon a mazy run into the box, before setting up Boyd for the shot but the new Rangers signing drilled just wide of the target.
Peter Lovenkrands then tried his luck from the edge of the area with similar results, before Nacho Novo tested goalkeeper Paul Mathers.
Peterhead were not completely without chances of their own and Bobby Linn was not too far away with a shot which ran across the face of goal and wide but goalkeeper Ronald Waterreus never looked troubled.
Only the heroics of Mathers prevented Rangers from taking the lead when Lovenkrands exchanged a superb one-two with Barry Ferguson only for his point-blank shot to be blocked by the Peterhead goalkeeper.
The Dane then saw a decent effort blocked at the near post by the impressive Mathers, who denied Boyd again moments later.
The breakthrough finally came in the 35th minute when Kyrgiakos grabbed the opener.
Novo met Lovenkrands' corner to direct a header at goal, but that was blocked by the body of Boyd, who then saw his own effort crash off the post, but Kyrgiakos reacted the quickest to stab home the rebound.
Boyd got his Rangers career off to the best possible start with a goal from the penalty spot after Lovenkrands was brought down.
That came five minutes after the break and he added a third four minutes later when he was on hand to convert a cross by Novo, following good work by the Spaniard.
Boyd's dream debut was complete when he headed in Alan Lowing's cross to seal a second-half hat-trick.
The goal deluge that had been threatened in the first half was at last coming to pass and McCormack scored Rangers' fifth with a powerful shot aided by a wicked deflection to take it past the unfortunate Mathers.
Rangers: Waterreus, Murray (Lowing 56), Andrews, Kyrgiakos, Smith, Novo, Malcolm (Alex Rae 72), Ferguson, Burke, Kris Boyd, Lovenkrands (McCormack 66).
Subs Not Used: Klos, Buffel.
Goals: Kyrgiakos 35, Kris Boyd 50 pen, 54, 71, McCormack 75.
Peterhead: Mathers, Hegarty, Tully, Perry, Gibson, Sharp (Robertson 89), Buchan, Hagen, Cameron, Linn (Youngson 77), Wood (Michie 72).
Subs Not Used: McCafferty, Calvin Shand.
Ref: B Winter