Even after the hosts saw Rory McAllister sent off midway through the second half, the visitors could not add to their advantage in what was an evenly-matched affair.
Missing captain Lee McCulloch through an ankle injury, the visitors made just one change to the side that had beaten Berwick Rangers handsomely last weekend, with Ian Black returning to the side in place of Dean Shiels, having completed a one-match suspension.
Peterhead are very much in the mix for the play-off places at the moment, and it was the hosts who were first to threaten. Dean Cowie won a free kick in the midfield and Neil Alexander in the Rangers goal did well to turn away a Scott Ross header from Ryan Strachan's delivery.
Rangers' first effort on goal saw Black seize upon Scott Ross' slack pass, to fire a snapshot from 25-yards, but his effort was always curling away from the target.
The goal then came the half-hour. Peterhead's Graeme Smith, himself a former Rangers goalkeeper, did well to save a Wallace shot, but could not hold on and Sandaza reacted first to fire the ball high into the net.
Peterhead hit back, and some fine hold-up play by McAllister created an opening for David Cox, but he dragged his effort wide of the post.
And the Blue Toon started the second half brightly, too, as Cox squeezed out Black on the touchline and cut inside, before flashing a left-footed effort over the crossbar.
The visitors, though, started to gain a territorial advantage, and came close when Lewis McLeod's cross found the head of Chris Hegarty at the back post, but his header failed to trouble Smith.
Peterhead's task was made much harder after 65 minutes when McAllister was adjudged to have handled in the area, and received a second booking, reducing his side to 10 men.
With a man advantage and Peterhead's discipline starting to go, the visitors began to take a firm grasp of the game.
Indeed, Templeton worked his way into the box with 20 minutes remaining, only to drag his low shot wide of Smith's far post.
Rangers could have sealed the points with five minutes remaining. McLeod 's clever reverse pass found Templeton in acres of space on the 18-yard line, but the former Hearts man sent his effort wide of the upright.
Peterhead were determined to battle for everything, though, and in injury time, Neil Alexander was forced to turn a Cox corner over his own crossbar, but Peterhead could not find an equaliser that few would have begrudged them.