Andy Little was also on the score-sheet in a match which saw Lee Wallace sent off in the second half and the visitors fail to level from the penalty spot.
The win allowed Rangers to move six points clear at the summit of the Irn-Bru Third Division.
Rangers chief executive Charles Green took to the pitch to address the crowd ahead of the match and confirm that a total of £22.2million had been raised through a share issue. However, while his speech was taking place, a section of the support staged a protest against potential plans to sell the stadium naming rights.
The hosts looked dangerous from the outset and Barrie McKay saw his shot blocked by goalkeeper Alex Mitchell, before Francisco Sandaza fired wide from the rebound. The pressure paid off after 29 minutes when Rangers surged into the lead thanks to a lovely low drive from Templeton from just outside the area that caught the opposition unaware.
Annan were handed the chance to haul themselves back into the match amid dramatic scenes in the second half.
Wallace was shown a straight red card by referee Paul Robertson, who ruled the full-back hauled down Jack Steele in the box, while skipper Lee McCulloch earned himself a booking for dissent.
Scott Chaplain stepped up for the spot-kick but goalkeeper Neil Alexander did well to smother with 60 minutes on the clock as Rangers retained their lead.
The home side then doubled their advantage after 64 minutes when Little raced onto a Templeton pass before sending an angled drive past Mitchell and into the far corner.
Templeton added to his own tally six minutes later with a low shot from 12 yards, shattering any hopes of an Annan comeback.